Radiation levels measured by the Apollo missions suggest a human absorbed dose of around 0.1 rad/day; crudely that's about 1mSv/day - about 1,000 times the background radiation level in a place with lots of igneous rocks, or about 10,000 times the dose from eating a banana. It would certainly look impressive in a cloud chamber, but it is not a damaging amount of radiation - it is about seven times the dose allowed for nuclear power plant and other radiation workers in the USA, which is set with a fair margin for safety; or about the same as the dose permitted for workers during an emergency.
For the Apollo missions of less than two weeks, that level of exposure was not considered particularly dangerous; of the 12 Apollo astronauts who walked on the Moon, eight are still alive today; James Irwin died of a heart attack at 61; Alan Shepherd died of Leukaemia at the age of 74 (27 years after his flight); Pete Conrad died in a motorcycle accident aged 69; and Neil Armstrong died at 82 from complications arising from coronary bypass surgery. This sample is really too small to say anything other than that exposure to 1mSv/day outside the van Allen belts for between one and two weeks doesn't cause any really obvious long term health effects; The medium term effects were negligible; And the short term effects were more interesting than they were dangerous - astronauts even in low earth orbits report seeing cosmic ray 'flashes' as the radiation lights up the inside of their eyeballs even with their eyes closed.
So the answer is that the activity in a cloud chamber outside the van Allen belts would be about 1,000 times the activity at the Earth's surface; which might look scary, but is probably harmless for exposures measured in days or weeks, rather than months or years.
