Does having the air conditioning on help kids with asthma?

In some ways. Asthma is an inflammatory disease aggravated by allergies, heat, and humidity. Air conditioning allows asthma sufferers to remain cool without having to open windows, allowing pollens and other irritants into the house. However if the irritants are already in the house, like cigarette smoke, dust or mold in the ductwork, etc, then open windows could be the better option.
Sometimes. If your child's triggers are environmental allergens, things like tree pollen, then keeping the windows closed and breathing filtered, air conditioned air can be of help in minimizing exposure to those triggers. If your child is triggered by things like dust mites or viruses or smoke, then probably not. Deal with dust mites with a mattress cover and don't smoke. And smoking outside doesn't help.