Ships can be cool inside, especially at night no matter where they are - even in the southern Caribbean. Some like to bring a light sweater or similar.
During the day shorts and t-shirts or similar things are the normal attire. At night shorts are okay as long as you are not going to the main dining room. If you want to eat at the buffet dinner you can wear shorts, but in the main dining room (or the extra-fee dining locations) the expectation is for collared shirts and pants for dinner service. Please note that you CAN wear shorts and t-shirts to the main dining room for breakfast and lunch.
On formal nights a jacket and tie is, again, the expectation but men will be allowed in without them. Many people choose to simply use the buffet dinner on formal nights. The buffet dinner is not to be dismissed as being cafeteria quality. They do a great job with it and usually have some of the entree items being served in the main dining room. We often will try the dinner buffet at least once, especially on those nights when we don't want to spend a lot of time in a dining room setting. We can set the pace and get on with the evenings agenda. If you do plan to use the buffet, please let your waiter know if you are using assigned dinner seating.