Luckily I don't really have to deal with anything about being a fur irl. One of my friends admitted to being a furry and explained to the possy what it is so they didn't get the wrong idea, then another admitted, and then me. So yeah.. I think they know, but they don't really care. One thing that does bother me irl is Vix, who likes to shout "YIFF!" at the top of her lungs in public, just to see who turns around. Like seriously.. shut up!
But if you wanna get rid of the stigma, don't mention it. It's not like it's of huge significance and everybody needs to know you're a fur. If you happen to wear a tail or something and people ask why, you can explain a bit about furry and escape the stigma. If that person does research and finds "Yiff in hell fur fag!" pages, just tell them it's a bunch of bored idiots who don't actually know what furry is. Sure, explain that a SMALL bit of what they say is true, some people (very few) do what the trolls say furries do, but they're just a small percentage of the fandom, if that.
Just do that, and I almost guarantee you that anybody who's not an idiot will take your word for it.