I'm not sure why you have an aversion to doing this correctly - either on the server a la
PrintMotd no
PrintLastLog no
and
#/etc/pam.d/ssh
# Print the message of the day upon successful login.
# session optional pam_motd.so
Or adding ~/.hushlogin for each user.
Hint, for ~/.hushlogin, add it to /etc/skel so new user home directories are created with the file.
Update:
Without more information about your backup cron job, my only other suggestion is to redirect the output of the command to a file (or let cron capture it in email) and the output of the ssh session to /dev/null. Something like:
0 0 * * * ssh backuphost "backup_script_that_writes_to_a_log" >/dev/null
Or
0 0 * * * ssh backuphost "backup_command 2>&1" >/dev/null
I'd have to play around with the commands a bit, but that should get you started.
Best Answer
You can't. (At least not with stock OpenSSH)
The server banner is sent by the server before authentication happens. It's point is usually to contain a legal disclaimer or similar "If you're not authorized disconnect NOW" type message, or other critical things you don't want the remote user to be able to suppress/ignore.
If you really want to get rid of this you will need to hack and compile your own customized version of the SSH client.