In general, they do not. If the radiation that is emitted is neutron radiation then it can transmute what is hist but most radioactive materials do not emit neutron radiation.
There is a danger of radioactive contamination but that is a physical transfer of material. So if there is radioactive dust it can be physically transferred to your clothes.
So your clothes do not charge there is just now material that is radioactive in them.
Compare to how if you stand in the smoke of a fire your clothes will smell of smoke. The smoke did not change the clothes but some of it got stuck in them.
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