Maybe I misunderstood the original question. If it is what I think it is, it will be Carnival employees checking you in for the cruise. Birth Certificate from the Dept of Vital Statistics as stated is fine for this along with Driver's License. The people that have been turned away recently have been those using BC's from hospitals etc. which are not acceptable. I have been cruising for nearly 20 years with my BC without any problems. And mine isn't even a conventional one, but a plastic credit card type thing. It has the magic words on it though. Issued by Virginia Department of Vital Statistics.
They don't check anything to get off the ship in ports. You simply "ping" out with your Sail & Sign card on the A-PASS machine. To get back on the ship you need your Driver's License and Sail & Sign card.
Now, if something unfortunate (missing the ship) were to happen in a foreign port a Passport Card would certainly help expedite matters it would seem to me. In fact I got one for this very purpose to carry off the ship.
Here is a link to Carnival's information.