First of all, your $query isn't really authenticating, since it's asking for the password instead of supplying it. This makes more sense to me:
SELECT username FROM clients WHERE username='$username' AND password='$password';
However, there are some security aspects that are relatively easy to take care of:
You could encrypt the password using the MySQL function PASSWORD():
SELECT PASSWORD('coffee');
Other than that, restrict the MySQL access privileges as much as you can (see chapter 6 in the MySQL manual), make sure MySQL requires a username and password to allow the script access to the DB, and give the PHP scripts as few access privileges as you can, so no-one but you can view the source code when logged in on FTP.
I'm not sure about the last one; I'm not sure if the scripts stop working if you remove the Read permission for 'nobody'.
I'm sure there are more things that can be done, but these are the things I know of. Keep your passwords safe.