Most places in the Caribbean do not switch to daylight savings time. They are far enough south that they don't need to.
I'm almost positive that the ship will change its clock to local time so that the ship's time and local time are the same. I say almost positive, because I haven't sailed on the western itinerary (obviously), but they did this on the eastern trip.
On the eastern route, the ship's clock moves forward an hour on Sunday night (I think, it might be Monday), so that it is on Atlantic Standard Time, which is what is used on St. Thomas and St. Maarten. On Thursday night, the clock moves back an hour, back to EST, the time zone for the Bahamas and Florida. The exception to this is during daylight savings time. Since EDT and AST are the same, the clock doesn't change, but the ship is still on local time at the ports of call.
In any case, they will make it very clear what the ship's time and local time are. When the clocks change, they leave a note in your stateroom in the evening with your navigators, as well as making announcements at the shows and over the ship's public address system.
