The passport book is the best option, but there are a couple of benefits to the cards over birth certificates, in my opinion.
First, the cards are more durable than paper copies of birth certificates; You can easily carry the cards to the beach or on excursions without having to worry about protecting paper from the elements.
Second, if there were some unusual reason that we had to interact with the local border patrol on an island, I feel more confident in presenting a passport than my birth certificate. Would they sort everything out with the birth certificate? ... I'm sure they would; but I bet it would go faster with a Passport (card or book).
All three methods are valid for closed-loop Caribbean cruises; you need to do what makes you comfortable.