For Caribbean ports, U.S. citizens need only show a photo ID such as a drivers license plus your cruise card. I don't recall needing anything more than my cruise card at Coco Cay since it is a private island. The ship will post ID requirements for the ports in the daily Cruise Compass (newsletter).
A color copy of the main two pages of a passport is a good idea. I know a lady who lost her passport in Europe. Since she had the photocopy to show at the Embassy they were able to expedite making her a new passport.