If it is a "closed-loop" cruise you do not need a passport. A closed-loop cruise is one that stops in North America (Canada, Mexico and The Caribbean) and returns to the same port it started.
Most round-trip cruises out of Miami, Fort Lauderdale, Tampa, New Orleans or Galveston are these types of cruises.
With a closed-loop cruise you will need an original or government issued copy of your birth certificate and a government issued picture ID (drivers license).
Having told you the requirement I still advocate that you get a passport. If, for some reason, you need to fly home from the Caribbean you need a passport. The birth certificate and drivers license are only good for sea travel. The passport also allows you to travel wherever you like.