It depends on when your cruise departs, and if it is considered "closed loop". Generally, on or after June 1st a passport (or passport card) is required. However, there is apparently an exception which you can see here
. I got a bit annoyed with this topic last week since I think people need to stop procrastinating and get a passport.
Bear in mind that even right now you can't fly back to the U.S. without a passport. So if you have to leave your cruise for some reason you will be facing a problem trying to get home.