Nancy is quite right. I would go a step further, though, and recommend you get a passport. You have plenty of time to do that, as the processing generally takes less than 8 weeks. And though it may seem expensive, a passport lasts 10 years. If you plan to do more traveling in the future - whether another cruise or on land - it will come in handy. There is also the chance that at some point, either the U.S. or some of the Caribbean countries may start requiring passport use for those traveling abroad, even on a cruise.
I personally find having a passport is very convenient. Rather than having to take special care of my original birth certificate, folding it carefully, making sure it doesn't tear, or worse - that I don't lose it, and rather than having to juggle two or more pieces of ID, the passport is all-in-one. It is accepted for boarding an airplane, even for travel within the U.S., boarding a cruise ship, or flying to another continent.