We have never had a problem when we used birth certificates on such cruises.
You may hear about how awful it is if something happens and you need to fly home and can't because you don't have a passport. Obviously you aren't stuck overseas forever (Americans abroad have their passports stolen all the time). You'll need to get an "limited validity emergency passport" through the nearest consulate or embassy. When I worked at an embassy years ago this was not the rare exception, it was something that happened more often than you think.
However, it is a hassle, so you roll the dice and hope you don't need to fly home for whatever reason. You almost certainly won't, but if you do the US is there to help. Nice feeling.
Sorry to provide more information than you asked (!), but I'm telling you this in case someone comes along and gives out partial information.