My understanding is that it is both fresh and recycled (but highly filtered) air.
Recycled airplane air isn't as disgusting as you think, writes Dave Levitan in Psychology Today. In fact, it's cleaner than the air in your office.
"Air on an airplane is about half fresh and half recirculated, and that whole volume is changed out several times per hour, pilot Patrick Smith, author of Ask the Pilot on Salon.com told Levitan. “Recirculated air passes through the same kind of high-efficiency particulate air filters hospital ORs use--which remove almost all bacteria and viruses."
I have travelled a bunch of times during the holiday season and also during March with my now 3.5 yr old. No major issues. I think the most likely thing you are anyone on a plane may get is a cold. I think the illness while when you fly connection has more to do with the contaminated surfaces (ie tray tables, armrests, bathrooms etc) than the air. Those surfaces are not sterilized between flights or regularly at all. Your best bet is to wash hands frequently or use lots of hand sanitizer on the flight and keep hands away from your face/mouth nose etc. That and boost your kids immune system for a few weeks prior to travel.