Remote Desktop Connection Without Domain Name

You can save credentials. If you save the connection as an RPD file, and use that file to initiate the connection, you can specify in options for the connection "Allow me to save credentials". It will prompt you the first time then shouldn't afterwards. It's possible your domain is set up in such a way that you can't save the credentials at all.

There are two things you should check:

1. Your domain user has to be in the local group Remote Desktop Users or Administrators(on the server)
2. Check the security policies on the server with Local Security Policy(secpol.msc). Make sure your user is not in Deny log on through Remote Desktop Services. You find this in Local Policies\User Rights Assignment

Usually you don't have to check the policy Allow log on through Remote Desktop Services because the Remote Desktop Users are already added to this one. Those policies could be managed through your domain which would not allow you to make any changes.