I love Belize, but beaches is not what Belize is famous for because there aren't too many. The nicer beaches are at Ambergris Caye and Placencia and those are way too far to get to on a day trip while on the cruise. You could probably make it to Ambergris by flying out of Belize City, but I think it would be best suited for a land trip.
Spanish Caye is nearby, it's a quick boat ride to and from. Hugh Parkey has a nice dive shop and restaurant on the other side of the island. There are several tour operators that sell tours from inside the tourist village and they will take you out there gladly. There is also a beach break on Bannister Island which is OK.
I personally love Caye Caulker which can be reached via water taxi, a short walk through terminal 4 and out the gates of the tourist village left a block or two. Once you reach this island there is an area at the split where people swim at times. There's also a terrific open air restaurant just down from where the water taxi stops called Rainbow. They have ice cold Belikin beer, and fresh yummy seafood. I have eaten there many times and it is delicious. I highly recommend the red snapper and the fresh conch. There is a tour provider there called Tsunami Adventures who will take you around the area if you like. It ain't fancy, but it is a place we love to go.
The primary touristy things to do for Belize cruisers is going swimming with the sharks and stingrays, ziplineing and cave tubing. I don't do the cave tubing or zip lineing because we cruise with our kids and its not something we or they are interested in. The sharks and Rays we loved and we've done it many times. We've used Coral Breeze and Marvin Deras Ecological tours. they both do a good job and they offer all of those tours.
The sharks (nurse sharks) are hard to find anymore at shark ray alley, but the stingrays are still there and plentiful. Do make sure to leave all your shiny jewelry back on the ship, because there is always one very hungry looking barracuda nearby and they like shiny stuff. They usually combine the shark and stingrays tour with a snorkeling tour of the Coral Gardens, which is very nice too if you like to snorkel.
Let me know if you need further info on any of this. I love Belize, but all the cool stuff is outside the tourist village and so you need to book it with a reputable tour company or do it on your own via water taxi.
If there are any divers in your group, I highly recommend Seasports Belize they are very close to the tourist village, they are owned by a couple of Marine Biologists and I had a great couple of dives with them last year.