I don't believe that any centralized records/statistics exist detailing how many people were rejected for each possible reason, so unfortunately most of what we have to go on is anecdotal. There are, however, anecdotes that back up the idea that it was used and even recommended as a way of avoiding the draft.

Two members of the Byrds mentioned claiming homosexuality (or at least being evasive about it in the draft interview) in Johnny Rogan's Requiem For The Timeless, a biography of the band:

The psychiatrist asked me, ‘Have you ever done it with a woman?’ I said, ‘No!’ He said, ‘Have you ever wanted to?’ I said, ‘No!’ He said, ‘Have you ever done it with a man?’ I said, ‘No!’ He was just looking at me — that’s all he was doing. Finally, he said, ‘Get the hell out of here.’ They ran me down the red line and booted my ass out of there — ‘Get out of here, you faggot bastard, schizoid homosexual, not fit for military service at any time — 4-F.’

Abbie Hoffman recommended it among many other methods of avoiding conscription in his classic Steal This Book in 1971:

Check off as many items as can't be verified when given the forms. Suicide, dizzy spells, bed-wetting, dope addiction, homosexuality, hepatitis. Be able to drop a few symptoms on the psychiatrist to back up your story of rejection by a cold and brutal society that was indifferent, from a domineering father that beat you, and mother that didn't understand anything. Be able to trace your history of bad family relationships, your taking to the streets at 15 and eventually your getting "hooked." Let him "pry" things out of you if possible. Show him your letter if you had the foresight to get one.

The book Chance and Circumstance: The Draft, the War, and the Vietnam Generation has an interview with a "draft counselor" who supposedly advised some of his clients to use this tactic successfully, although apparently one guy who really was gay ended up getting drafted anyway.

One thing to remember is that at the time, homosexuality was still down in the DSM as a mental illness. If you look at Hoffman's list, most of the other things he says to claim fall under mental illness, and he even goes so far as to advise faking a mental breakdown and getting admitted to a psychiatric hospital in the section on draft evasion. If you were trying to get out because of a mental illness, documentation might have been requested to prove you really had whatever affliction you claimed. Of course, as Hoffman also points out, being gay is not as easily proven or disproven as other medical issues, which is why he recommends it over other possible tactics.

EDIT: A comment that will no doubt get deleted very soon mentioned Arlo Guthrie's 1967 song "Alice's Restaurant Massacree," very loosely based on a true story of Guthrie's arrest for littering and subsequent rejection from the draft because of that arrest. Two lines stand out referring to homosexuality. One is a more comedic line about "father rapers" with obvious homosexual connotations, but mostly played for laughs:

And I, I walked over to the, to the bench there, and there is, Group W's where they put you if you may not be moral enough to join the army after committing your special crime. And there was all kinds of mean nasty ugly looking people on the bench there.

Mother rapers. Father stabbers.

Father rapers! Father rapers sitting right there on the bench next to me!

And they was mean and nasty and ugly and horrible crime-type guys sitting on the bench next to me. And the meanest, ugliest, nastiest one, the meanest father raper of them all, was coming over to me and he was mean and ugly and nasty and horrible and all kind of things and he sat down next to me

Again, that's more a joke based on the absurdity of the situation. But at the end of the song, he makes a more pointed remark about resisting the draft, again referencing homosexuality:

And friends, somewhere in Washington enshrined in some little folder, is a study in black and white of my fingerprints. And the only reason I'm singing you this song now is cause you may know somebody in a similar situation, or you may be in a similar situation, and if you're in a situation like that there's only one thing you can do and that's walk into the shrink wherever you are, just walk in say "Shrink, you can get anything you want, at Alice's restaurant." And walk out.

You know, if one person, just one person does it they may think he's really sick and they won't take him. And if two people, two people do it, in harmony they may think they're both faggots and they won't take either of them. And three people do it, three, can you imagine, three people walking in singing a bar of Alice's Restaurant and walking out. They may think it's an Organization. And can you, can you imagine fifty people a day, I said fifty people a day walking in singin a bar of Alice's Restaurant and walking out. And friends they may thinks it's a movement