Airports are shared infrastructure, just like roads. As such they are serving a common purpose, and don't have to make money by themselves (rather indirectly through enabling business). That's exactly where the government should be involved in. E.g. It makes more sense to have a single airport rather than several competing ones, airports shouldn't go bancrupt and be closed, and building them might require changing laws or evicting/relocating 1000s of people. These are all in the sphere of public responsibility.