Neutrons make things radioactive because they impact the nuclei and change it to something unstable (radioactive).
Not that neutrons are a component of direct cosmic radiation in any case -- having a mean lifetime of 15 minutes or so. They would be produced by spallation when primary cosmic rays impact the heavy nuclei in shielding. They're nasty buggers really, as their Q weighting is as high or higher at all energies as protons and alpha particles (from 5 to 20), and much greater than gamma rays and electrons (1). Even when thermalised, their Q is 5. The most effective shielding for neutrons contains lots of protons, so water, paraffin (kerosene) wax, graphite, or polythene are typically used. For other types of radiation, you need high atomic number and high density materials for shielding -- ie, lead.