The plastic in the drawer units is not tight. I had a lot of that storage and found some carpet beetles (they eat organic stuff- major infestation all over town one year) in drawers with drawing supplies. Ink is NOT organic and neither are lettering pens! If they can get in, so can air..
The separate bins and Artbin cases are tight. We live in a dry area, so no problem.
If damp is a problem, you can use cheap fine cat litter in a sock. When it gets wet, put it in the oven, dry it out and reuse it.
Dry air, static electricity and cat hair are a major problem. I can't run a small humidifier since the house is open plan.
I prefer dry air when sleeping.