Well, lets see how many people yell at me for this:
Last June we were there at the beginning of "Gay Days". I was really not worried about it, even though I have 2 young children. I take that back...I was worried that coming from RED-NECK-VILLE USA, my younger son would LOUDLY ask a "gay" couple questions, and make them feel uncomfortable.
But to answer your question...It was nothing. You should be more worried about the crowd than anything. The wait at MK is at beyound peak...so you might want to plan on going to MGM, or AK that day...Epcot seemed pretty crowed too. Most of the gay day partisipants do realize that this is a family theme park, and that many people are uncomfortable with their lifestyle, and are very considerate. They are usually there with their families, and are wanting to just enjoy themselves too.
Yes, most of the group wore red, but so did I so don't base your entire day trying to avoid red shirts.
Like I said this did not really bother me...But the things that do bother me are the tween's & teens that run you & your kids over to beat you in line, cus and "make out" in line. Had a young punk...maybe 19, hit his girl friend in line right in front of me...of course I called him out on it. Within minutes the two had made up and were going at it hot and heavy...explain that to your kids...

(dang that made me sound old, didn't it)
Anyway, just try to make the most out of you vacation and have fun. Just decide what you want to tell your children ahead of time if they do see something and ask you about it!