Nothing. Okay, that’s not exactly true, but anything that might happen to air travel isn’t a big deal.
Most federal government employees related to air travel are “essential”.
During a government shutdown, “essential” workers would continue to work. This includes air traffic controllers, TSA, and Customs and Border Protection. Planes aren’t going to start falling out of the sky, and you’re not going to be stuck in the immigration hall for weeks. SOME TSA checkpoints and Customs and Border Protection stations may be staffed lighter during a government shutdown, so security and immigration lines might be longer, but that can be solved with Global Entry and Pre Check.
If you haven’t applied for a passport and need one within the next couple of months, it might be a good idea to go to a passport office in person and apply with expedited service. Turnaround times can be as little as a few days. If you’re applying for a passport for a trip more than three months or so out, apply for your passport by mail, but do it soon. Assuming that passports are processed in the order that they are received, you want to be early in the queue when passport process restarts following shutdown. I would expect that routine passport service will jump by several weeks, should there be a government shutdown.
Of course, all of this is assuming that the government will get shut down, but if it does, having your passport in-hand and being a bit patient is the best preparation you can get.