I don't think the passport card is a good idea -- it's only valid for entry to the US from "Canada, Mexico, the Caribbean, and Bermuda at land border crossings or sea ports-of-entry".
The reason to have a passport on a cruise is in case something goes wrong -- for example you get left behind and have to fly from one island to another (in another country). Doing so without a passport is way more stressful. Or say you have to fly home.
But the passport card won't get you there, and it won't get you home by plane either. Very limited usefulness, probably only for people living near the Mexico or Canadian border.
So if you want the "safety" of a passport, get the full book. Otherwise, go with your birth certificate and make sure you get back to port way ahead of schedule.