It depends on how much you want to spend. If money is no object, I would recommend a cruise on Star Clippers:
http://www.starclippers.com/index_en.html
It is a romantic (good for the honeymoon) ship , going under sail, unlike your standard cruise line. It is much more laid back that a typical cruise, and lacks all the "vegas glamour", which we found nice when we went.
Best of all, there is a dive shop in the stern of the ship, where a platform folds down from the back of the ship. From this platform, they launch the zodiac(s) to take a nice dive right outta the ship you are sailing on.
The nice thing about this is that they take you to many destinations around the carribean, and there is a ton of variety of things to see ranging from great shallow reefs, to 130 ft deep wrecks, or nice sandy bottoms.
My second vote would be to go out of Boynton Beach, Fl. They offer some really nice drift dives over some great reefs. I would personally take Boynton over the Keys any day, although once in that area you could take a drive down to the Keys for some variety.