I have done 3 cruises in the last 3 years, all in January. The Bahamas sea can still be a bit on the cool side (mainly in the morning, afternoon will be fine for us), but once you get to the Caribbean zone it was nice. We were at Grand Cayman early January this year and day time high was around 84F. But as PP mentioned, because we were flying in from Ontario Canada, we had to deal with the more likely uncertainty of the flight, and have to fly at least one day prior, if not more.
We did Eastern Caribbean in early August a few years back and it was hot and humid, but was tolerable because it was already very hot and humid at Miami and getting further down to San Juan was just a bit more humid. I thoink it was about 103F at Labadee.
We will try cruising the Caribbean in late October next year.