Most alcoholic drinks are, in fact, nonflammable because they're mostly water.
The original definition of "proof" was that 100-proof was the minimum alcohol content required for a pellet of gunpowder soaked in the drink to still be able to burn. That corresponds to about 57% ABV, which is considerably stronger than most spirits. (The current definition of proof used in the US is that 100-proof means 50% ABV; the rest of the world, as far as I'm aware, has discarded proof and just uses ABV.) So, unless your alcohol is extremely strong, you can't even burn gunpowder in it. Beer would be about as good as water for putting out fires; probably wine, too.
You're probably thinking that alcoholic drinks are flammable because of their use in cooking. However, the reason you see impressive sheets of flame when a chef pours alcohol into a pan is that the pan is very hot. The heat instantly boils the alcohol and the chef then sets fire to the vapour above the pan. If, instead, they allowed the dish to cool to room temperature, then added the alcohol and dropped a match in the pan, the match would go out.
The real danger of selling alcohol in airports is people getting drunk. People do get obnoxiously drunk in airports and on flights and it relatively often causes problems – every few weeks you hear about a flight being diverted to deal with a violently drunk passenger. But, at the moment, it's felt that it's not a big enough problem to come close to banning the sale of alcohol.
You can buy lighters in airports because you're already allowed to carry them with you through security. Most things on a plane are reasonably well fire-proofed so somebody setting fire to their clothes or something like that can probably be dealt with by dumping a fire-extinguisher on then, before the fire spreads. And anybody lighting a lighter on a plane will attract immediate attention from everybody around them, followed by the cabin crew, followed by the police after landing.