If you created a secret like this (just as example):

kubectl create secret generic mysql-pass --from-literal=password=YOUR_PASSWORD

then use it as environment variable:

 spec:
      containers:
      - image: mysql:5.6
        name: mysql
        env:
        - name: MYSQL_ROOT_PASSWORD
          valueFrom:
            secretKeyRef:
              name: mysql-pass
              key: password

or mount it as volume:

spec:
  containers:
  - name: mycontainer
    image: mysql
    volumeMounts:
    - name: foo
      mountPath: /etc/foo
      readOnly: true
  volumes:
  - name: foo
    secret:
      secretName: mysql-pass
      items:
      - key: password
        path: my-secret/my-password

password secret is stored under /etc/foo/my-secret/my-password file instead of /etc/foo/password.

Reference: https://kubernetes.io/docs/tutorials/stateful-application/mysql-wordpress-persistent-volume/#create-a-secret-for-mysql-password

EDIT:

I made a try with the file you linked in your answer and I've seen that passwords missed in a couple of points.

Here the file I succesfully used:

apiVersion: apps/v1beta1
kind: StatefulSet
metadata:
  name: mysql
spec:
  selector:
    matchLabels:
      app: mysql
  serviceName: mysql
  replicas: 1
  template:
    metadata:
      labels:
        app: mysql
    spec:
      initContainers:
      - name: init-mysql
        image: mysql:5.7
        command:
        - bash
        - "-c"
        - |
          set -ex
          # Generate mysql server-id from pod ordinal index.
          [[ `hostname` =~ -([0-9]+)$ ]] || exit 1
          ordinal=${BASH_REMATCH[1]}
          echo [mysqld] > /mnt/conf.d/server-id.cnf
          # Add an offset to avoid reserved server-id=0 value.
          echo server-id=$((100 + $ordinal)) >> /mnt/conf.d/server-id.cnf
          # Copy appropriate conf.d files from config-map to emptyDir.
          if [[ $ordinal -eq 0 ]]; then
            cp /mnt/config-map/master.cnf /mnt/conf.d/
          else
            cp /mnt/config-map/slave.cnf /mnt/conf.d/
          fi
        volumeMounts:
        - name: conf
          mountPath: /mnt/conf.d
        - name: config-map
          mountPath: /mnt/config-map
      - name: clone-mysql
        image: gcr.io/google-samples/xtrabackup:1.0
        command:
        - bash
        - "-c"
        - |
          set -ex
          # Skip the clone if data already exists.
          [[ -d /var/lib/mysql/mysql ]] && exit 0
          # Skip the clone on master (ordinal index 0).
          [[ `hostname` =~ -([0-9]+)$ ]] || exit 1
          ordinal=${BASH_REMATCH[1]}
          [[ $ordinal -eq 0 ]] && exit 0
          # Clone data from previous peer.
          ncat --recv-only mysql-$(($ordinal-1)).mysql 3307 | xbstream -x -C /var/lib/mysql
          # Prepare the backup.
          xtrabackup --prepare --target-dir=/var/lib/mysql
        volumeMounts:
        - name: data
          mountPath: /var/lib/mysql
          subPath: mysql
        - name: conf
          mountPath: /etc/mysql/conf.d
      containers:
      - name: mysql
        image: mysql:5.7
        env:
        - name: MYSQL_ROOT_PASSWORD
          valueFrom:
            secretKeyRef:
              name: mysql-pass
              key: MYSQL_ROOT_PASSWORD
        ports:
        - name: mysql
          containerPort: 3306
        volumeMounts:
        - name: data
          mountPath: /var/lib/mysql
          subPath: mysql
        - name: conf
          mountPath: /etc/mysql/conf.d
        resources:
          requests:
            cpu: 500m
            memory: 1Gi
        livenessProbe:
          exec:
            command:
            - bash
            - "-c"
            - |
              set -ex
              mysqladmin -uroot -p$MYSQL_ROOT_PASSWORD ping &> /dev/null
          initialDelaySeconds: 30
          periodSeconds: 10
          timeoutSeconds: 5
        readinessProbe:
          exec:
            # Check we can execute queries over TCP (skip-networking is off).
            command:
            - bash
            - "-c"
            - |
              set -ex
              mysql -h 127.0.0.1 -uroot -p$MYSQL_ROOT_PASSWORD -e "SELECT 1" &> /dev/null
          initialDelaySeconds: 5
          periodSeconds: 2
          timeoutSeconds: 1
      - name: xtrabackup
        image: gcr.io/google-samples/xtrabackup:1.0
        ports:
        - name: xtrabackup
          containerPort: 3307
        command:
        - bash
        - "-c"
        - |
          set -ex
          cd /var/lib/mysql

          # Determine binlog position of cloned data, if any.
          if [[ -f xtrabackup_slave_info ]]; then
            # XtraBackup already generated a partial "CHANGE MASTER TO" query
            # because we're cloning from an existing slave.
            mv xtrabackup_slave_info change_master_to.sql.in
            # Ignore xtrabackup_binlog_info in this case (it's useless).
            rm -f xtrabackup_binlog_info
          elif [[ -f xtrabackup_binlog_info ]]; then
            # We're cloning directly from master. Parse binlog position.
            [[ `cat xtrabackup_binlog_info` =~ ^(.*?)[[:space:]]+(.*?)$ ]] || exit 1
            rm xtrabackup_binlog_info
            echo "CHANGE MASTER TO MASTER_LOG_FILE='${BASH_REMATCH[1]}',\
                  MASTER_LOG_POS=${BASH_REMATCH[2]}" > change_master_to.sql.in
          fi

          # Check if we need to complete a clone by starting replication.
          if [[ -f change_master_to.sql.in ]]; then
            echo "Waiting for mysqld to be ready (accepting connections)"
            until mysql -h 127.0.0.1 -uroot -p$MYSQL_ROOT_PASSWORD -e "SELECT 1"; do sleep 1; done

            echo "Initializing replication from clone position"
            # In case of container restart, attempt this at-most-once.
            mv change_master_to.sql.in change_master_to.sql.orig
            mysql -h 127.0.0.1 -uroot -p$MYSQL_ROOT_PASSWORD <<EOF
          $(<change_master_to.sql.orig),
            MASTER_HOST='mysql-0.mysql',
            MASTER_USER='root',
            MASTER_PASSWORD=$MYSQL_ROOT_PASSWORD,
            MASTER_CONNECT_RETRY=10;
          START SLAVE;
          EOF
          fi

          # Start a server to send backups when requested by peers.
          exec ncat --listen --keep-open --send-only --max-conns=1 3307 -c \
            "xtrabackup --backup --slave-info --stream=xbstream --host=127.0.0.1 --user=root"
        volumeMounts:
        - name: data
          mountPath: /var/lib/mysql
          subPath: mysql
        - name: conf
          mountPath: /etc/mysql/conf.d
        resources:
          requests:
            cpu: 100m
            memory: 100Mi
      volumes:
      - name: conf
        emptyDir: {}
      - name: config-map
        configMap:
          name: mysql
  volumeClaimTemplates:
    - metadata:
        name: data
      spec:
        accessModes: ["ReadWriteOnce"]
        resources:
          requests:
            storage: 10Gi

Remember that you need a config-map:

apiVersion: v1
kind: ConfigMap
metadata:
  name: mysql
  labels:
    app: mysql
data:
  master.cnf: |
    # Apply this config only on the master.
    [mysqld]
    log-bin
  slave.cnf: |
    # Apply this config only on slaves.
    [mysqld]
    super-read-only

the services (used inside the scripts:

# Headless service for stable DNS entries of StatefulSet members.
apiVersion: v1
kind: Service
metadata:
  name: mysql
  labels:
    app: mysql
spec:
  ports:
  - name: mysql
    port: 3306
  clusterIP: None
  selector:
    app: mysql
---
# Client service for connecting to any MySQL instance for reads.
# For writes, you must instead connect to the master: mysql-0.mysql.
apiVersion: v1
kind: Service
metadata:
  name: mysql-read
  labels:
    app: mysql
spec:
  ports:
  - name: mysql
    port: 3306
  selector:
    app: mysql

and a persistence volume to be used by statefulSet.

The answer provided by Nicola Ben is close, but there were a few mistakes in the StatefulSet declaration.

The main problem is that the environment variable is declared in the mysql image but not in the xtrabackup where it is also referenced. Secondly the START SLAVE and xtrabackup commands should take the user and password as well.

Here is a working version of the StatefulSet yaml declaration:

apiVersion: apps/v1
kind: StatefulSet
metadata:
  name: mysql
spec:
  selector:
    matchLabels:
      app: mysql
  serviceName: mysql
  replicas: 2
  template:
    metadata:
      labels:
        app: mysql
    spec:
      initContainers:
      - name: init-mysql
        image: mysql:5.7
        command:
        - bash
        - "-c"
        - |
          set -ex
          # Generate mysql server-id from pod ordinal index.
          [[ `hostname` =~ -([0-9]+)$ ]] || exit 1
          ordinal=${BASH_REMATCH[1]}
          echo [mysqld] > /mnt/conf.d/server-id.cnf
          # Add an offset to avoid reserved server-id=0 value.
          echo server-id=$((100 + $ordinal)) >> /mnt/conf.d/server-id.cnf
          # Copy appropriate conf.d files from config-map to emptyDir.
          if [[ $ordinal -eq 0 ]]; then
            cp /mnt/config-map/master.cnf /mnt/conf.d/
          else
            cp /mnt/config-map/slave.cnf /mnt/conf.d/
          fi
        volumeMounts:
        - name: conf
          mountPath: /mnt/conf.d
        - name: config-map
          mountPath: /mnt/config-map
      - name: clone-mysql
        image: gcr.io/google-samples/xtrabackup:1.0
        command:
        - bash
        - "-c"
        - |
          set -ex
          # Skip the clone if data already exists.
          [[ -d /var/lib/mysql/mysql ]] && exit 0
          # Skip the clone on master (ordinal index 0).
          [[ `hostname` =~ -([0-9]+)$ ]] || exit 1
          ordinal=${BASH_REMATCH[1]}
          [[ $ordinal -eq 0 ]] && exit 0
          # Clone data from previous peer.
          ncat --recv-only mysql-$(($ordinal-1)).mysql 3307 | xbstream -x -C /var/lib/mysql
          # Prepare the backup.
          xtrabackup --prepare --target-dir=/var/lib/mysql
        volumeMounts:
        - name: data
          mountPath: /var/lib/mysql
          subPath: mysql
        - name: conf
          mountPath: /etc/mysql/conf.d
      containers:
      - name: mysql
        image: mysql:5.7
        env:
        - name: MYSQL_ROOT_PASSWORD
          valueFrom:
            secretKeyRef:
              name: mysql-pass-root
              key: password
        - name: MYSQL_USER
          value: server
        - name: MYSQL_PASSWORD
          valueFrom:
            secretKeyRef:
              name: mysql-pass-user
              key: password
        - name: MYSQL_DATABASE
          value: medlor
        ports:
        - name: mysql
          containerPort: 3306
        volumeMounts:
        - name: data
          mountPath: /var/lib/mysql
          subPath: mysql
        - name: conf
          mountPath: /etc/mysql/conf.d
        resources:
          requests:
            cpu: 500m
            memory: 1Gi
        livenessProbe:
          exec:
            command: ["mysqladmin", "-uroot", "-p$MYSQL_ROOT_PASSWORD", "ping"]
          initialDelaySeconds: 30
          periodSeconds: 10
          timeoutSeconds: 5
        readinessProbe:
          exec:
            # Check we can execute queries over TCP (skip-networking is off).
            command:
            - /bin/sh
            - -ec
            - >-
              mysql -h127.0.0.1 -uroot -p$MYSQL_ROOT_PASSWORD -e'SELECT 1'
          initialDelaySeconds: 5
          periodSeconds: 2
          timeoutSeconds: 1
      - name: xtrabackup
        image: gcr.io/google-samples/xtrabackup:1.0
        env:
        - name: MYSQL_ROOT_PASSWORD
          valueFrom:
            secretKeyRef:
              name: mysql-pass-root
              key: password
        ports:
        - name: xtrabackup
          containerPort: 3307
        command:
        - bash
        - "-c"
        - |
          set -ex
          cd /var/lib/mysql

          # Determine binlog position of cloned data, if any.
          if [[ -f xtrabackup_slave_info ]]; then
            # XtraBackup already generated a partial "CHANGE MASTER TO" query
            # because we're cloning from an existing slave.
            mv xtrabackup_slave_info change_master_to.sql.in
            # Ignore xtrabackup_binlog_info in this case (it's useless).
            rm -f xtrabackup_binlog_info
          elif [[ -f xtrabackup_binlog_info ]]; then
            # We're cloning directly from master. Parse binlog position.
            [[ `cat xtrabackup_binlog_info` =~ ^(.*?)[[:space:]]+(.*?)$ ]] || exit 1
            rm xtrabackup_binlog_info
            echo "CHANGE MASTER TO MASTER_LOG_FILE='${BASH_REMATCH[1]}',\
                  MASTER_LOG_POS=${BASH_REMATCH[2]}" > change_master_to.sql.in
          fi

          # Check if we need to complete a clone by starting replication.
          if [[ -f change_master_to.sql.in ]]; then
            echo "Waiting for mysqld to be ready (accepting connections)"
            until mysql -h 127.0.0.1 -uroot -p$MYSQL_ROOT_PASSWORD -e "SELECT 1"; do sleep 1; done

            echo "Initializing replication from clone position"
            # In case of container restart, attempt this at-most-once.
            mv change_master_to.sql.in change_master_to.sql.orig
            mysql -h 127.0.0.1 -uroot -p$MYSQL_ROOT_PASSWORD <<EOF
          $(<change_master_to.sql.orig),
            MASTER_HOST='mysql-0.mysql',
            MASTER_USER='root',
            MASTER_PASSWORD='$MYSQL_ROOT_PASSWORD',
            MASTER_CONNECT_RETRY=10;
          START SLAVE USER='root' PASSWORD='$MYSQL_ROOT_PASSWORD';
          EOF
          fi

          # Start a server to send backups when requested by peers.
          exec ncat --listen --keep-open --send-only --max-conns=1 3307 -c \
            "xtrabackup --backup --slave-info --stream=xbstream --host=127.0.0.1 --user=root --password=$MYSQL_ROOT_PASSWORD"
        volumeMounts:
        - name: data
          mountPath: /var/lib/mysql
          subPath: mysql
        - name: conf
          mountPath: /etc/mysql/conf.d
        resources:
          requests:
            cpu: 100m
            memory: 100Mi
      volumes:
      - name: conf
        emptyDir: {}
      - name: config-map
        configMap:
          name: mysql
  volumeClaimTemplates:
  - metadata:
      name: data
    spec:
      accessModes: ["ReadWriteOnce"]
      resources:
        requests:
          storage: 10Gi

I also noticed that in his file he has the replicas set to 1 which would only start the master and no slaves, so that might explain why some were getting startup failure on the slaves.