I raised my children with cloth diapers. Airing out the baby back then meant loosely pinning or laying the diaper over the important area, and leaving the plastic pants off so that air can circulate around the skin during nap time. My daughter said that since she didn't have any cloth diapers, she had just held the baby with nothing on at all. The urine went all over the place.
If you don't have cloth diapers, the answer to this problem is in the kitchen. Use a kitchen towel. Choose an old one. As long as it's clean, it will do the job. If the baby doesn't roll over, just lay it on the important parts. If the baby rolls in its sleep, use diaper pins to keep it on. If it gets soiled, swish it in the toilet to remove the solids, and then wash it with the rest of your whites.
I know this because when I had children in diapers, and I had one of those days when I didn't get all of the laundry done, and I ran out of clean diapers, I would (oops, don't tell) bundle their bottoms in a kitchen towel instead. Today, kids run to the store. Back then, we had to be more creative.
I bought some kitchen towels for my daughter that look just like diapers. They are woven cotton cloth without all the terrycloth loops. I thought they would make good burping cloths. I used to use cloth diapers to protect my shirt, too.
Towels come in handy during potty training, as well. When my two-year old son had a problem with wetting the bed during nap time, I stopped using a mattress pad, and used a beach towel under the fitted sheet instead. Mattress pads have to be hung out on the clothes line to dry. Beach towels can go in the dryer. I used the old ones. No one complained.
Research has proven that cloth diapers are better for potty training because the wet feeling is motivation to learn. A little extra laundry now means you solve the problem faster.
Disposable diapers have their assets, but they can't do all of the things that a cloth diaper can do. So, in my opinion, all mothers need a basket of kitchen towels and a package of diaper pins to take care of the problems that disposables can't solve.