I too am going to Belize for the first time at the end of May. I don't know about the affect of cruise ship people, but I do have an idea of what days...belive it or not, I do cruise. But even on a cruise, I try to avoid going where the masses of other cruise ship people go when doing shore excursions!
If you go to a online cruise travel site, sometimes you can select a cruise based on port of call. If you select "Belize City", it will bring up the ships that call there. I did this on a cruise travel web site for May, and it seems that Carnival, Royal Caribbean, and Norweigan all have ships going to Belize City Tue-Fri. From frequenting the cruise message boards, cave tubing at Caves Branch, snorkeling at Shark Ray alley an going to Coral Beach (wherever that is) seem to be really popular destinations. So is going to Altun Ha. If you decide to do a day trip to any of these locations, be prepared.