If I remember rightly,
DCL will give us a photocopy at check-in.
However, I always carry a photocopy of our passports whenever we travel - separate from the originals, of course. And I keep a photocopy at home so that if necessary I could always ask someone to come into the house, find it and fax it.
Our credit card insurance policy (the one where you just phone one number and they cancel all your cards if lost or stolen) allows you to store a copy of your passport online so you can access it from wherever you are. But we're too paranoid to use that facility.
Even if we were allowed to keep our passports, I wouldn't carry them, they'd be in the safe for the duration of the cruise. We always carry another form of identification to get back onboard.