They recommend a color copy of the page with your picture (it needs to have your passport number, signature, and picture on it). You can use a photo ID to get back on the boat at the ports unless the ship specifically tells you you need your passport (we needed to do that when we were in Spain or maybe it was Portugual--can't remember, but we all stood in a really really long line and had to go through basically airport security screening with X-rays, etc. Lots of armed guards--made me a little nervous since we've never had to do that procedure getting back on the ship at a port. But it was all good in the end).
Our family got the passport cards when we renewed/got our kids passports so when we go into Canada we wouldn't have to drag the book (we're just a couple of hours from BC and like to visit occasionally). Also, it serves as ID for when our children fly--I have very tall children and have had issues with TSA in the past making a "to-do" about my kids not taking their shoes off to go through X-ray scanners, etc. Now we have proof of their age. But we actually all just got our Nexus cards which basically makes traveling with those passport cards redundant (so, waste of $$) because they have photo id on them. It was a sweet deal: for $50/per adult (kids under 18 free), we get TSA pre-check, global entry, and can use the Nexus lanes at the Canadian borders (I think Mexico has special lanes we can use too). So if you live close to the Canadian border, would be worth the lengthy application process! Whew...off the tangent of passport/Nexus cards!