How to auto start MySQL if that stop for any reason?

Digital Ocean has a tutorial about it, which also includes an example on how to keep MySQL running. The short synopsis is:

1. Install monit

   sudo apt-get install monit
   

2. Edit the configuration file:

   sudo nano /etc/monit/monitrc
   

   and include the following in it:

   check process mysqld with pidfile /var/run/mysqld/mysqld.pid
       start program = "/etc/init.d/mysql start"
       stop program = "/etc/init.d/mysql stop"
   

3. Reload the new configuration:

   monit reload
   

I strongly suggest you look at the MySQL error log to understand why it is stopping (it should not do this in normal operation). It would be a good idea to also run mysqlrepair to check the status of all tables as you might have some serious DB corruption.