November to March seems to be the best time as you will (hopefully) miss the hurricane season.
Which cruise line to select depends very much on your personal preferences. Do you like to dress up in the evenings? If so Cunard takes some beating in terms of ambience, service and quality of food. If you prefer a more casual aproach, then you may well like one of the American ships e.g. Princess or Celebrity. My best advice would be to have the best grade cabin you can afford and go for a balcony cabin which is especially pleasant in the warm weather of the Caribbean. I agree with Mr/Mrs Miller on the question of the best time to travel.