We just returned from our first cruise on Grand Princess. We climbed Dunns Falls in Ocho Rios and it was awesome! My husband and I purchased river shoes before we went and they were terrific. Ours were from Soloman - a little expensive but well worth it. They are made like slip on tennis shoes, but made for the water. The soles were nice and thick and very comfortable. Our friends wore thinner soles water shoes and said they could feel the rocks on their feet. You definately need something good because the falls are very slippery in spots. I'd make the purchase before you go - they will be more expensive on the islands.
Hope you have a great time!