So, apparently if you let tap water sit for a few days the chlorine will evaporate off but how can one safely expose the water to the the air without risk of infection from airborne particles?
Letting the water sit to let the chlorine evaporate will not make a difference when it comes time to brew. Once the wort starts to boil, it will kill anything that got in there.
So you don't have to wait overnight for the chlorine to leave, just boil your water for about 20 minutes. If you don't even want to wait that long, just dissolve a campden tablet in your water before you start to use it. This will strip all the chlorine related chemicals.