If it is a closed loop cruise that begins and ends in the same US port and if they are a US citizen that was born a citizen and not naturalized then they can use a birth certificate and government ID combo to leave and enter the US on a cruise, but not by air. Since 2009 both Jamaica and Grand Cayman require a passport to enter their country. You'll have to check with the cruise line, but they should be able to stay on the ship. Here are the locations that can process a passport if you are leaving within 2 weeks:
http://travel.state.gov/content/passports/english/passports/information/where-to-apply/agencies.html
Have them get a real passport and not a passport card at the info sheet for Jamaica says they may not accept the cards.