US Immigration requires all cruise passengers to present themselves to an INS official before anybody is allowed off the ship. It doesn't matter whether you would like to stay on board or not. Every single passenger has to report to the INS official, otherwise the ship won't be cleared.
A letter will be delivered to your cabin prior to arriving in St. Thomas and Miami, telling you at what time to present yourself to US Immigration.
I understand an original birth certificate is still sufficient, but - as a "foreigner" - I'm not 100% sure.