Radioactive particles (dust) in air are problematic. The gas in air does not become radioactive (at least in a dangerous way) from exposure to radiation.

If the surface of mars is exposed to radiation capable of making particles become radioactive, does that mean the surface is already radioactive? Or would this only happen to airborne dust particles, not to sand particles on the ground?

The dust itself would have to be something highly radioactive like uranium. Radiated silica isn't going to get into you and cause real issues.

Couldn't irradiated silica dust get into your lungs though?

If you're breathing silica dust you're going to get silicosis anyway, so there's not much point worrying about it.

Maybe the living area buildings inside the tent can have a vestible / "dustlock" for keeping the dust outside, and outside those it just becomes standard to wear a dust mask.

Dust is one thing but it’s the fines that really get you.