In Debian, I setup
nodm, such that my computer logs in automatically when it is turned on. I am not concerned about thieves accessing the computer directly, but compared to a setup using GDM, can such a setup make the system less secure through the Internet?
The autologin as such shouldn't be any more insecure than logging in manually - you would need to trust anyone with physical access to the computer, but again, that's the same issue no matter how you start the X session.
However, whatever applications you are running on login may be a security issue - e.g. if you were to do something like "xhost +" in your setup, that would allow others to connect to your running X session with your privileges, which is generally a bad thing.
You should, of course, also setup firewall, use the autologin with an account with low privileges, require password when using sudo, etc etc... but all that falls within recommended security practices regardless of how you login.
Unless you use autologin together with XDMCP, the security implications are the same as leaving an unattended X server running without any screen locking mechanism active.