I think you can do a lot with the single ship premise, but I believe the key to it is to have a variety of different stories. Both the original Enterprise
and the Enterprise
-D weren't always
out exploring--they sometimes were back in Federation space dealing with things there, enabling them to avoid the "alien of the week" syndrome and mix things up, even show other Starfleet ships, facilities, and personnel on occasion.
Exploration is fine as a premise, but it's really just an excuse for our heroes to be out there so they can have lots of different (non-exploration) adventures as well.
I also think you also have to have a strong set of characters that audiences can relate to (or root for/hate to love), as well as be able to drive a number of stories as well.