Brian,
We've done cruises like that in February for the last few years. Mostly on Celebrity. Who are you going with? If it's one of the big lines they'll have a golf program that'll arrange the games for you at each stop (where there's a course). You can find info about the program and the courses through the cruise line web site.
I assume you'll be leaving from San Juan, so there'll be no time for golf. If there is time they'll probably take you to Conquistador. Long hike east of San Juan. Interesting mountain type course overlooking the ocean.
At La Romana they'll usually offer you either Teeth of the Dog or Dye Fore. The former is better by a lot. Trip to course takes 5 minutes.
St Thomas has one course, Mahogany Run. It was in much better shape when we were there last year. Plays up and down a canyon. Some interesting holes. The par 3 on the ocean cliff-top has to be seen and played.
Dominica - don't know, never been there.
St. Kitts - a decent course renovated recently by a Canadian architect, Tom McBrrom. They used paspalum greass - interesting if you haven't played it before. The course has some neat holes on a relatively flat piece of land. The closing holes are along the ocean. It has its own web site you can google for.
Barbados - have done the Barbados Golf Club - a pedestrian effort, but affordable. Last time there we were taken to the Country Club course at Sandy Lane. Nice design (by Fazio
), fairly hilly, nice condition. More expensive. I think the World Cup was there last year.
All in all, I'd disagree with Gary, the courses are decent to good and sure as hell beat shopping or touring in any of these places. Especially if you're coming from a cold weather place.