This has changed a little over the years and cruise passengers should carry some form of photo ID with them (I would advise them to, anyway) - BUT, in the Caribbean at least, all we (and that means me because I am responsible for all such things) ever bring ashore off a ship are the color photocopies of our passports and that has ALWAYS been enough (often not requested), even with the guys at the pier in St Thomas (are they TSA? - don't rember - but they do check ID for sure in my experience). In addition to carrying a color photocopy of your passport, you also should always have access to a copy of your passport online - not because of getting back on the ship, but just in case something bad happens and you miss the sailing of the ship.