I agree, keep them where air can move around them, I use one of those hand carry baskets from the supermarket. Keep them in the dark, cool, dry, and no where near your onions.
Store onions the same way, but not where they share the air with the potatoes.
You will be surprised how the problems disappear. I keep mine under the kitchen cabinets in two totally separated areas.
Oh my, I never knew I was supposed to keep onions and potatoes separate. Since at this very moment they are in a not so cool tub sharing space, I will:
1. Put them in an empty clean bike basket.
2.Put it in a cooler place.
3.Segregate the onions from the potatoes.
http://www.fruitsandveggiesmatter.gov/month/onion.html
“Onions should be kept in a cool, dry open space away from bright light. Onions do best in an area that allows for air circulation.
Because onions absorb moisture, do not store onions below the sink. Also, do not place onions near potatoes because potatoes give off moisture and produce a gas that causes onions to spoil more quickly.”