In one of its senses, the word airy means not having much substance (i.e., of an argument or statement) or showing lack of concern. In this sense of the word, “fairy” is often added on to the end of airy to make it airy-fairy, as in “his airy-fairy ideas.”
I don’t know if airy-fairy originated as a slur against gays. I’m guessing so because they’ve been the victims of many slurs over the years. This is likely one of them.
However, I don’t think it’s intended that way in most cases today. It’s just something people say, with its derogatory origins, if that was its origins, long since forgotten.
Often, airy-fairy is used by people on the right wing of the political spectrum as an insult against what they see as flaky ideas of people on the left wing of the political spectrum. (Flaky generally means that the ideas are formed after considering all aspects.) Then again, a number of people who are socially conservative are somewhat homophobic, so who knows what they have in mind?
Even when the phrase “airy-fairy” is clearly not intended as a slight against gays, some people might still take offense, so act accordingly. I firmly believe that people deserve of our respect regardless of their race, creed, sex or sexual orientation.
Well, maybe not all people. As far as I’m concerned, it is perfectly just and good to treat Nazis and neo-Nazis disrespectfully with extreme prejudice.
In another sense of the word, airy means well ventilated. This is generally considered to be a good thing in a temperate climate, but it can be a bit of a bother in the winter in a cold climate or in any climate when a hurricane, cyclone or tornado blows through.
Finally, airy can also mean light and graceful in movement. Ballerinas, for example, could be called airy. Professional ballerinas who are incapable of giving a performance that could reasonably be referred to as airy might be called unemployed.