SSH makes all typed passwords visible when command is provided as an argument to the SSH command

If you provide a remote command to run, SSH doesn't allocate a tty, so the remote command is unable to disable echo. You can force SSH to provide a tty using the -t option:

ssh -t user@server 'mysql -u user -p'

The equivalent option (for -o or for config file) is RequestTTY. I'd caution against using it in config because it can have unwanted effects for non-interactive commands.

Storing the password in a protected option file

If you can trust ^[*] the remote computer's security, you can store the password in a properly protected option file, as suggested in the End-User Guidelines for Password Security chapter of the manual, without the need to communicate through ssh or to type each time.

Specifically you can add a line in the [client] section of the file .my.cnf in your home directory:

[client]
password=your_pass

Of course you have to keep that file from being accessible to anyone but yourself, by setting the file access mode to 400 or 600 with, e.g.

chmod 600 ~/.my.cnf

Then you can use something like

ssh user@server 'mysql -u user110971 --defaults-file=/home/user110971/mysql-opts'

where user110971 is the username of your account.

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Forcing the ssh to allocate a pseudo tty (ssh -t)

This problem happens each time that you send a command through ssh and you need to insert the input because, by default, ssh will not allocate a pseudo-tty.

You can force the tty allocation with the option -t, (even more than one if needed):

-t

Force pseudo-tty allocation. This can be used to execute arbitrary screen-based programs on a remote machine, which can be very useful, e.g. when implementing menu services. Multiple -t options force tty allocation, even if ssh has no local tty.

As you can read in this Debian post (Jul_11_2008) about sudo, it is an old issue that loves to recur:

ssh user@server "sudo ls"  
password: password  

And password is shown you

The solution is to force ssh to allocate a pseudo-tty, with the -t flag:

ssh -t user@server sudo ls

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Note:

^[*] If you can rely on leaving the password in a file accessible only by you and root on the working client.
If it is possible to reboot the remote computer changing OS or to remove the HDD, the computer can't be considered completely secure... but in that case the database itself will not be secure.

Mount option "hidepid" for proc fs is also valuable. It makes your commandlines invisible in process list for other users. fstab example:

proc /proc proc hidepid=1 0 0