Is the OP asking specifically about our views on LGBT people, or characters in video games?
If the former, I'm of the generation where discrimination based on race, geneder or sexual orientation is frowned on. It still happens but anyone of any faith, race, creed or whathaveyou is promised by law equal treatment and I wholly concur. I have a bisexual friend and 3 gay friends, 1 man, 2 women and don't think any differently of them for it.
I will admit I have an issue with the T in LGBT tho. I don't even really understand why they are included in a group with the others because transexuality is a vastly different set of issues altogether. I can't say I truly understand transexualism and I don't particularly agree with it though I would happily fight to defend their rights to do as they wish and be treated equally regardless. I don't know, nor have I ever known anyone transgender and I think I would find it awkward to deal with them...I consider it odd and aberant (the dictionary definition of the word), though perhaps no more so than previous generations considered homosexuality. The issue is mine, tho I don't feel the need to address it.
If we're talking characters in games, I'm not really sure that it's of import. BioWare, who others have mentioned, haven't really addressed the issue of homosexuality in games. It's simply treated that the PCs gender is irrelevant for commencing a romance with a character, whose dialogue, behaviour and storyarc doesn't change one whit (except for maybe "him"/"her" as appropriate). I don't think they necessarily have a place in video games...what I mean is that *unless* homosexuality is an issue important to the game's story or character development, it's pointless being there.
In most games (without romances) sexuality is irrelevant. Our FPS protagonists with neither voices nor legs would be no different whether they were gay, or otherwise, thus it is a needless detail that does nothing for the game. If it is an important detail, then I don't know what kind of game it is that makes such a point of sexuality, and regardless, I don't think it would be something I'd play. Not because of homophobia, but because I probably wouldn't enjoy a game like it, nor could I empathise with (or project onto) the protagonist.
I'd be curious to hear however, from anyone gay of either gender, would you gain more enjoyment from a game in which say the damsel in distress is instead a dude in distress, or the protagnist is a female and the damsel remains a damsel? I don't mean specifically a Mario/Peach type affair, but any similar story where currently the male protagonist and female plot-device might be substitued for a same-sex equivalent.