Few things are a sure as to start an argument as the role of the air conditioner in mileage. This article at AutoBlog talks about the issue somewhat.
The bottom line is, yes, the air conditioner uses a lot of power and will reduce your mileage, sometimes by as much s 9 miles per gallon. On the other hand, according to the AutoBlog article, rolling down your windows at high speed produces a lot of drag, which also reduces mileage.
A lot of it boils down to your priorities. If you place a high value on mileage, you should keep your windows up and your air conditioner off and sweat a lot. On the other hand, if you want to drive in comfort on a hot day, then turn the air conditioner off and be prepared to pay a little more at the pump. There may be a safety factor to the air conditioning: If you're hot and uncomfortable while driving, then that's a distraction.
According to this GasBuddy article, keeping the windows open at 45 mph and below is a more fuel efficient way to cool off than using the air conditioner. Then again, GasBuddy places a high priority on saving gas.
Judgment is also useful. Some people turn the air conditioner on at every whipstitch, even if it's just a little warm outside. At low speeds, driving with the windows down doesn't create a lot of drag and doesn't affect mileage. If it's not really hot, why not drive with the windows down, stay comfortable, improve mileage and reduce weear and tear on the air conditioner?
The air conditioner is there to keep you comfortable, so use it. Just use it mindfully.