You can buy a waterproof case that hangs on your neck to hold your credit card and ID card. You can buy them on the ship or ahead of time at any sport or dive shop. If you have a passport you want to carry, there are larger waterproof wallets. I've used both kinds of containers and both work well.
On the other hand, I've been on ship excursions and just left all of my stuff in a backpack on board and nothing has been touched.
You do not need to carry your passport into port with you. All you need is your ships id. Lock your passport in your stateroom safe. A lot of folks do carry a copy of the first page of their passport with them, just in case.
Don't bring a lot of cash, just enough for cab fare, a few small souveniors and maybe a quick lunch. Credit card all big ticket item purchases.
IF you are going to a beach, rent a locker.
Your items are safe on excursions, just put your wallet in your backpack. The tour companies depend upon the cruise ships business to keep them going. If they had a problem with theft they would be out of business in a flash.
Bring an underwater camera for snorkeling. Leave the expensive 35mm or the digital in your stateroom, in the safe.