I'm not creating an argument in any way with the poster who told you to stay on the ship in Belize but to say to stay on the ship because of thieves, is in my opinion, a little much.
There's all kinds of tours to Mayan Ruins if you're interested in that sort of history, etc. If not, there's plenty of other tours that you can book through the ship that will get you out and about and see some of the countryside.
If not that, then just ride the tender ( about a 12-15 minute ride over to the dock ) where the shopping complex which is modern and clean, is located. It's a secure area, numerous shops, 3-4 bars, ( The Wet Lizard comes to mind ) and is safe. I wouldn't go out roaming the streets and down the back alleys wearing flashy jewelry but I wouldn't do that here at home either.
I've been to Belize 5-6 times and never had a problem. However, just recently here in my own home a drywall worker stole about $30,000. in jewelry right out from under our noses. He's in jail now, our jewelry pawned all over hell and back so you don't have to go abroad to find "thieves ".
We have more than our share here in the U.S.
Take your cruise and just use common sense. To cruise and not get off the ship may be doing yourself an injustice, especially because your heard there were thieves somewhere- yes, there's thieves, here, there and everywhere. Heck--we even have them in Washington , D.C. and we pay them to be there !!
Go and enjoy your trip.