I would leave my passports in the room safe, and take a photo copy of the photo page of your passport with you.
I take a photo ID, photo copy of passport photo page, a bit of cash, and a credit card. There's two real scenarios here:
If the whole thing is stolen:
it's a little inconvenient...you can get back to the ship and prove yourself to get back on...a little more hastle to get back on...but nothing too outrageous, but then your passport is safe for you to use to get back into the US, and get on a plane to get back home.
Then you probably would need to put an ID alert on your credit reports and get your photo ID replaced...but you've got your passport to use as photo ID until you can get that replaced.
All told, you're out maybe $50 for the ID replacement plus however much cash you had...all told, not too big of a deal.
If you miss the ship:
This is MUCH more inconvenient and expensive...don't do this. I repeat: Do. Not. Do. This. If you do miss the ship, then you'll have your photo ID, a credit card, some cash, and all of your passport info with you. It's everything you need to get yourself to a US consulate, get a replacement passport, get a hotel (if needed), and get a flight booked to catch up to the ship (hopefully) and/or at least get home.
The costs here will be great, and there are no guarantees that you will be able to or allowed to rejoin the ship. There are many laws/regulations about this and you may not be allowed to rejoin the ship at a later port.
If you can't rejoin the ship, you'll have to get yourself a flight home and arrangements with the cruise line to get your stuff from the ship.
The costs here can be pretty significant, but
travel insurance will likely help depending on the reason(s) you missed the ship....but you should check your travel insurance as a lot of times they have exceptions for things that could be foreseeable, like traffic jams, and such.