Read the cruise ship's daily newsletter. Most major cruise ships produce a daily newsletter for guests. Newsletters include announcements of upcoming events, which often are based on the destinations. For example, a Caribbean cruise with a stop in Aruba will have an instructional seminar about the island before passengers visit Aruba. Seminars will include information on the geography of the island, the natives, shopping, food and shore excursions. Pick out activities on the newsletter that appeal to you and attend the seminars.
Try the daily drink special. Most cruise ships sailing the Caribbean offer a themed drink of the day, often served in a souvenir glass. Carnival, for example, has a drink called the Caribbean Dream. Royal Caribbean offers the Caribbean Cooler.
Enjoy the Caribbean music. Many major cruise lines bring the island atmosphere to the lido deck. Listen to live reggae music or steel drum bands while relaxing poolside.
Protect yourself from the heat and sun. Temperatures in the Caribbean vary from season to season and island to island, but even when it's coldest (in January) they generally don't fall below the mid-60s. Temperatures can reach the high-80s during the hottest months of the summer. The UV index in the Caribbean often is rated high or very high, so the potential for sunburn is increased.
Stick a medicated patch behind your ear to prevent motion sickness. According to the Mayo Clinic, scopolamine, available by prescription, can prevent seasickness without causing drowsiness, which is a common side-effect of oral over-the-counter motion sickness medications, including Dramamine. Generally, the Caribbean Sea is calmer than the open waters of the Atlantic Ocean, so seasickness might be less of an issue. If you are prone to motion sickness, though, get a prescription for the patch before you head out to sea--just in case.