Royal Caribbean and Celebrity sail from Bayonne, NJ.
The other poster is wrong. The policy varies by cruise line, but Royal Caribbean requires someone in the cabin to be at least 21. On Carnival, one passenger must be at least 25. The only exceptions are for couples under that age who are married, but proof of marriage is required. Not sure about Celebrity, but it should be easy to find on their website.
The rules are kind of irritating if you're in the restricted age range, but spring breakers gone wild are very unpopular with other cruise passengers.
I believe there is a port in nj. and i'm pretty sure you can go to the bahamas from there. as long as your all 18+ you will be fine. Might just want to check on line go to royal Caribbean you'll be able to see ports of call and the destinations they go to.