1. Tell everyone that you're about to miss your flight.
Tell your cabbie. Tell the people you're waiting behind at check-in. Tell the TSA agent in line. You'd be surprised how kind your fellow travelers can be if you explain that you're late, and that you're just looking for kindness/sympathy/the fastest cab ride to JFK in recorded history. Maybe it's your fault that you're late for the flight, or maybe it's the airline's fault. Either way, there's no harm in asking for a little help.
2. Find the one airline representative who can help you.
These are the people with superpowers at the airport. If you're running late for your initial flight, find an agent from the airline and ask them if they can escort you to the front of the security line.
If you're on a connecting flight that's running late, ask the flight attendants if they can make an announcement to the plane asking that other passengers wait until those with tight connections disembark first. This works especially well in cities that are known for their friendliness — like Salt Lake or Minneapolis — but passengers may be less cooperative elsewhere. (cough New York cough)
I've even seen pilots call air traffic control and request that a connecting flight be held a few extra minutes for passengers.
If you ask, they may be able to help, and they can save you time in the process.
If your flight gets cancelled, or you've got weather-related delays, then the airlines will let you rebook your flight for free. Just download the airline's app and log in with your frequent flier information so you can make changes on the go. It'll be much faster than waiting in line or calling the airline.
3. Be prepared to run.
If you're trying to make that connecting flight and you have to run, remember to make as much noise as possible. Seriously. No one's used to seeing a grown man or woman sprinting through an airport. But as long as you're yelling, "Sorry! Excuse me!" people will step out of the way. If you don't, you're going to accidentally run over someone's grandma.
And take advantage of the magazines in the seat pocket on the flight. They've got maps of major terminals, and you'll be able to plot out the right route to your gate. In some cases, you may have to take a train between terminals, and knowing the route there can save you a little time.
4. Be nice
Above all else, you'll have the best chance to make your flight if you're just nice to the people you need help from. If you're trying to get through security and the airline agent can't help you, walk to the first class line and show the TSA agent your ticket. If you ask nicely, they'll probably let you through.
If you know you're cutting it close on a connecting flight, make a joke to the flight attendants as you're boarding the plane — or just smile and be friendly. They'll remember you when you're asking for help later, and that one nice gesture might come back to you in a big way.