If you miss the ship (but don’t, just don’t—be back an hour before all aboard),
DCL will retrieve your passport from the room safe and leave them with the ship’s agent.
Conversely, if your passport is stolen or lost while onshore, you will have a mountain of trouble to deal with to get home—it took us a month to get home when we were kids and mom’s bag with the passports was stolen when I was a kid. And every time we traveled internationally after than, we had to go through additional screening/verification which sometimes took hours because our identities were flagged in the system as people who had our passports stolen.
You can check on the DCL site if a passport is needed to go on the cruise. If not, then you do not need to take your passport onshore at any of the ports. If a passport is required for the cruise, some ports may require them ashore and others might not. You can mention the ports here and we can tell you.