Why would anyone bother to send monkeys up that high? We can send instruments that can accurately measure radiation and get actually data, rather than “monkey died,” “monkey threw up and lost hair”, “monkey seems fine but developed leukemia 6 months later”, and other useless datapoints. It’s also a lot cheaper to send the instruments. They don’t need life support.
Executive summary: Apollo used trajectories above and below the highest-radiation areas of the belts, and traversed them pretty quickly. All of the astronauts wore dosimeters. The total radiation received was in the 0.5 rad neighborhood, except for Apollo 14, which received 1.15 rad. Nasa's tolerable limit was 400 rads whole-body and 50 rads to blood-forming organs.
Still, a certain amount of luck was involved. A very strong solar storm could have caused problems or even killed the astronauts.