Very well. They were and are exposed to radiation. What does it prove? That Astronauts have a much higher occupational radiation hazard? Even if they were exposed to intense space radiation, I fail to get how that somehow disproves the moon landing. It's not like being exposed to unfiltered x-rays will automatically cause you to keel over and die. Airline pilots who spend regular amounts of time in thin atmosphere (i.e. significantly reduced electromagnetic filtering) have the same occupational hazards, although to a lesser degree.
I bet if you were to do an anolysis of long-term health of astronauts, you'd find their medical records align with the increased exposure to radiation. But increased exposure to radiation does not automatically mean health problems. It means increased chances of health problems. It certainly doesn't mean automatic fatality.
