They're gone within two weeks, so I generally don't have that problem (I'm part Scots-Irish, I can eat spuds for dessert). The main thing is to make sure they are in a cool, dry and ventilated place that is very dark and that they are not just dumped in a drawer.
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.