What is legally required to board the ship and what is smart traveling practices are two different things. I have first hand experience needing to obtain a passport in a foreign country. It was NOT fun.
I had a client's wife fly down to Costa Rica with our team. She was an American citizen. Her passport was expired and she did not realize it. They let her in to Costa Rica, but when we were all at the airport getting on our flight home, they denied her boarding. I stayed back with her because I was the travel director and I spoke Spanish. I had to contact the embassy which was located in San Jose. We were in Gunacaste. It was an EXTREMELY expensive 4 hour cab ride to the embassy. Of course it was a Friday afternoon, so we had to pay for hotel nights in San Jose, Friday, Saturday, Sunday. Finally, on Monday, the Embassy opened and we were able to go in and get assistance for an expedited passport...mind you, she already had one...it was just expired, so less paperwork. Finally, we had the passport on Wednesday afternoon. The next flight was Friday from San Jose. We had the super expensive cab ride, the 7 nights of extra hotel rooms, the new plane tickets, the cost of the missed flight and the cost of all the expedite fees. To make matters worse, she missed a week of work and her employer considered it job abandonment so they fired her!
It was stressful and expensive getting her a new passport in a foreign country. After experiencing this with her, I vowed to NEVER travel without my passport again. I cannot imagine how much more stressful it would have been had it been some type of medical emergency. What if someone falls ill and misses the ship because they are in the hospital or if they are medevac'd out, then becomes stable enough to be transported back to the States where the medical care is significantly better than 3rd world countries, but now has to wait it out for a passport while the rest of the family continues on the ship because they did not have passports either? Or what if a traveler simply misses the ship and needs to meet the ship at the next port? No flying for you!! Its just not worth it in my opinion to travel without one, whether it is required by law or not, because circumstances change.
FWIW, our 2015 WBPC cruise dropped a passenger (and family) who fell ill off the shores of Guatemala. The boat anchored, a tug boat came out and met the ship and then took the family to a hospital in Guatemala. It was the closest medical facility, so that is where they went. Given, we had to have passports so the family was good there, but still, when its a medical situation, they take you to the closest care facility...passports or not.