Don't most countries require that visitors have passports on their person? Only US citizens could leave the ship.

On the cruises I've been on (granted, they were all 5+ years ago), they held my passport until the end of the cruise. I was hoping I could have got it back and tried to get some visa stamps in it (a little hobby of mine). They were firm on that I could not have it. I should have brought one of my other passports to try and get it stamped.

I am a UK citizen and have been on several cruises originating in the USA. They take my passport and keep it during the cruise. I then take my "stub" to pick it up I believe the day/night before we get off.

I am a non-US citizen and last year I was on a cruise that departed from USA to the Caribbean. The ship was Italian, but most of the tourists were from the United States, the vast rest Europeans. Nobody ever took my passport during the journey and I had it always with me. Before boarding the ship, still in the port, they copied its first page for the immigration "pre-clear" or how you Americans call it... Despite we had a safe in our cabin, they advised us to keep the passports always with us and especially when leaving the ship. As far as I remember, actually you couldn't board the ship without the passport with the ship's ID card issued to each passenger, or the driving licence if you were American (probably because they can travel on the Caribbean without it...afaik). The most interesting thing was that citizens of my country need visas for entering most of the countries I visited on the cruise (which I didn't have) and I didn't have problems at all... Very strange which I still don't understand.
In the ports nobody asked for the passports, it was only for boarding the ship (asked by ship's security) and for emergencies. I think it is more wise to keep the passport with yourself. If you lose it you always have a copy on the ship, or even with yourself in the bag or another pocket.

The wife and I cruise quite a bit, and always leave the passports locked on the ship. I also travel a lot and usually leave the passport locked at he hotel.
I do keep a copy of my passport with me and also have a copy of the passport and drivers licence uploaded to my hotmail account as a password protected pdf file.

Passport While Cruising
 

What I always do is make copy's of your travel documents, keep the original in the cabin safebox and carry a copy + original driver license for excursions.
But.... if for some reason you mis the boot and have to travel to the next port
if that port stay in a other country, then travel by airplane you can forget it there is no way. Although sometimes you can be lucky and apply a travel document for a few day's from customs on the airport. If you decide to carry the original documents and your afraid someone will stolen or lose then I would carry all the travel docs in a little bag on a cord around your neck and in your shirt.

By the way for Europian in Europe are a driver license or ID card enough as travel doc.