Unfortunately, while we bring them home from the store in plastic bags, storing them in the plastic bags leads to mold and sprouting. They need to live in a dry, breathable container, but I also hate buying baskets! Thankfully, sturdy paper grocery bags provide a great solution.
Just cut them off an inch or two below the upper fold. Two of them side-by-side fit nicely in my standard-size cabinet, so I can sort red and yellow onions. Trim a smooth curve in the front face for easy access to the onions.
The bags are sturdy enough that I can tug them out and shove them around while full of onions without fear of breakage, and the onions breathe well enough I haven't had any more problems with sprouting or mold.
I originally intended these to be extremely short-term storage solutions, but they're still working beautifully several months down the road. I'm not terribly worried about the aesthetics of my onion bins, so I'll stick with this reuse project until something better presents itself.