Are you talking about the mysql daemon (started at boot) or the mysql client?
Most daemons (background services) listen to the network. If they do so on a privileged port, then they have to be run as root to allocate the port, but they usually switch by themselves to a non-root user later on.
So the first place to check would be MySql's configuration to see if it does this already.
If it doesn't or if you want to run any process as a non-root user, you could do (as root):
su -c "the_command" non_root_user
(don't forget to enclose the commands in single or double quotes).
Or, if the program is to be run as the same non-root user all the time, you could chown' it so that the intended non-root user is the owner of the executable and then enable its setUID bit. This approach is however
not recommended and must be used with caution (for security reasons).