A rough idea of your dates would probably yield better advice. Are you trying to go in December? If so, you may be facing a long, uphill battle.
Just in terms of frequency, I'd say NAS would be better than most. You also have partner options. You could get an island hopper from there for pretty cheap to anywhere else in the Bahamas.
Is SJU considered Caribbean? That can't be too tough. I would also try SDQ.
Also, open your mind to Central America. Not quite as popular of a destination for our friends in the Northeast. SJO is fairly open up front.