The Western Caribbean stays pretty warm all year, being isolated in the Gulf. The eastern caribbean gets more trade winds that can keep it a tad cooler, but the humidity is high all year 'round. In all honesty, it is always hot once you get below Miami and the temperatures really do not vary by more than 15 degrees all year round (24 hours/day) from the low 70s to high 80s.
In all sincerity, I would think about a cruise from New York that goes to Port Canaveral and Nassau. Or Alaska or a summertime cruise to Canada/New England.
To answer your question, however, the coolest months are right now, Dec - Feb.