Ok, receiving mails has nothing to do with saslauthd.
Yes, of course
Just wanted to point out, that something
What's in /etc/courier/authmysqlrc? Is the MySQL password correct in that file?
Yes, the password was right.
I found a (1) space after the password
After starting the services again, I receive emails!
So all is working fine!
Thank you all for leading me to the right spot