I know that it's possible to make a particular forum only visible to a set of defined users but is it possible to make a forum only visible to registered users in general. In other words, users who aren't logged in can't view the contents of a particular forum.

From the manual:

So you can try something like:

Here's a topic with some solutions and codes you could try out for your purpose. Let us know if something of this helped:

Guests should not be able to even view the forums / Disable viewing of /hide forums unless logged in

The idea behind it is to manipulate with $clForums / $clForumsUsers arrays on the fly — the code provided on the second page of the topic above adds any logged user to the allowed array, at the time unlogged users don't see the closed forum at all.

Yes, that's a better and complete solution from Paul's link.