Merlin Missy has been active in online fandom since 1994. She likes fanfics with plots and happy endings. (Note: All pronouns in the following are set to "she." If you are offended by this, you are being too sensitive and need to get over it. See? First lesson already learned.)
First, when someone in your fandom points out that something that occurred in your show's canon can be construed as racist, sexist, homophobic, or otherwise Not Cool, start by telling the observer she's being too sensitive, that she's reading too much into the material, and that she needs to remember this is just entertainment. It's fantasy / science fiction / horror. If she continues, especially if she points out a pattern of problematic material in your series, you should tell her she's just looking for things to be offended about. (This is true! People are always looking for things to be offended about. You can tell because people are always getting offended about something, especially if those things have always been that way. Since you never noticed anything was wrong, it's probably all in her head. Or their heads.) Also, you should try the "Get a life" routine, because that always kills at parties.
If more fans agree with her, you'll need to get more active. Point out that you or someone you know / love / met on a plane once are / is a member of the group she says is being put down. (Remember, if you have a friend who's not white, you can't be a racist. If you're male and love your mother, you can't be a sexist, and women never ever support or contribute to misogyny. Come on, try it!) Tell her you're not offended and/or the person you know isn't offended, therefore it's not actually offensive. If she shows you a list of stereotypes that have been shown by your series, make sure you claim that a) you've never heard of those before (so she must be making them up) and/or b) you don't see that at all in the material (so she's trying to create trouble by pretending there's a problem when there isn't one). Also come up with a good list of excuses why it doesn’t count in this case. Examples: "The character in question isn’t human." "It was supposed to be a joke." The latter is especially good, because you can also accuse her of lacking a sense of humor.
Attack her vocabulary and choice of words. If she didn't bring it up earlier, attack her for being a hypocrite now, and ignore her if she tries to claim she's been worried about this for a while and/or has brought it up among friends. If she's ever said something that you (or someone else) might find offensive, use that against her as a reason she has no leg to stand on now. Bring up tangentially-related things that could also be found offensive, point out that she's never complained about those (or that no one ever has, if you can conveniently claim later that you had no idea) and again, use it to prove this particular example can't be so bad. If she's using any kind of metaphor in her posts, make sure you interpret it to mean she's trying to incite an overthrow of the show and completely change everything about it. The word "militant" is always good to use here. You definitely want to paint her as an extremist in her opinion, and you're the one trying to take the sane middle ground.
Be sure to bring up any example you can scrounge up of an exception to her argument. If she's complaining about misogyny, find a female character who hasn't been murdered, raped, marginalized, forgotten or lobotomized. (For some series, this may require a bit of digging, so be patient. Finding that one exception to rub in her face will feel so good!) If she's calling the series on racist undertones, focus on one episode where terrible things didn't happen to the characters of color, or better yet, point out that bad things happen to some white characters too. (Ignore her "percentages" nonsense. If five non-white characters were killed, and five white characters were killed, what does it matter if there were only five non-white characters on the entire series? We want absolutes!) Homophobia? Well, that's just being true to the character's nature.
Whatever you do, don't research. Don't learn about coded sexist language. Do not verse yourself in racist stereotypes. Don't become familiar with the current statistics about violence against women or homosexuals, and be sure to claim that, while there may have been some racism in the past, violence against people of color simply doesn't happen anymore and shouldn't be relevant. Claim to be colorblind (a handy phrase invented in the '70s by anti-affirmative action folks) so that you don't even notice things like the race of a character. Claim that it's only realistic for so many women to be abused in fiction, and that you have always been a strong supporter of realism in your favorite series about muscle-bound, time-travelling rednecks from outer space who work with dragons and fight zombies for a living.
Make sure you claim that events, costumes, and so on are equally exploitative. For example, if you're a superhero fan, point out that it's exactly the same thing to show a fully-clothed man in spandex showing off his muscles as it is to have a close-up on a woman in a strapless bathing suit, emphasis on the folds at her crotch. If two characters of color are shown in a knock-out, drag-down brawl while the other (white) characters are standing around, cheering and betting on the outcome, point out that white characters have punched people before, so it's exactly the same.
Whine that you're being oppressed. This is important! The other fan and her friends are coming from a place of centuries of wrongs piled on wrongs, of attitudes so deeply ingrained in Western society that it takes a backhoe just to dig them up in the first place, much less start setting them aside. They might possibly have the moral high ground here. You cannot allow this to happen. You must, and I cannot stress this enough, claim that your opinion is not being valued. You must whine that although your verbal opponent is discussing a case of intolerance in the series, she is being intolerant of your right not to have your squee harshed. You are the victim here! You are the one being hurt! Your right to enjoy your show without having to think at all about the deeper social meanings of the stories unfolding before you on the screen is being jeopardized! All you wanted was some mindless entertainment and the opportunity to meta about it for hours with your friends online. You didn't want to hear about why it's unsettling to see three female characters of color end up in maid uniforms in one season. It's just not fair. Your freedom of speech is being limited. (Um. You'll have to dig for that one on your own, but I'm sure you can come up with something.)
Make it all about you. We're fans, we're good at that. The other fan and her friends will likely be using "I" constructs. ("I feel this is a problem." "I'm really bothered by the recurrence of this particular trope.") Use your "I" statements wisely to say that everyone has an opinion and is entitled *cough* to that opinion. You can agree to disagree, as long as nothing at all changes on the series, right?
Never ever under any circumstances admit you're wrong. Even if she can cite chapter and verse about the skeevy fate of every character of color on the show. Even if she proves beyond a doubt that what you thought was a joke was actually rape. Even if the character in question, in the very next episode, starts quoting Adolf Hitler (Godwin! Hooray!) and says s/he fully supports the Final Solution. It's still just an opinion, and one come to by interpreting the text from the wrong perspective by a humorless militant. You're just disagreeing.
Finally, defriend everyone who disagrees with you, who said something elsewhere unflattering about your position, and/or who has the wrong character in her icons list. Those people aren't your friends anyway and you're better off without them stinking up your Friends list. Start up a new community all about the squee. If you happen to see her or her friends be happy about a later episode, be sure not to point out any racism or misogyny that you see in it for the purposes of actual discussion, but do ask why the "feminists / radicals / pick your word here but be sure they know you mean them" aren't up in arms over XYZ. When they explain why XYZ isn't offensive, either pick a fight over their previous arguments (even better if this takes place more than a year later) or do your very best to have zero understanding of what the difference is between an episode where a black lesbian is captured and assaulted versus an episode where she is briefly captured, then frees herself and her friends, and single-handedly beats up the villain. (This should be easy. If you've come this far on your journey, you probably really don't get it anyway.)
Congratulations! You are now officially an Internet jerk! Your prize is the much-coveted Golden Boot Award. Guess where you can shove it?
(For the purposes of not offending her regular readers, Doctor Merlin assumes those who regularly visit her column understand the nature and purpose of over-the-top satire. Doctor Merlin and Firefox News do not endorse or condone the shoving of actual boots, golden or otherwise, into any orifice. Doctor Merlin does endorse and condone self-education, and she strongly encourages the twin paths of "trying to do better next time" and "be excellent to one another".)
Edit: Please go visit Livia's lovely follow-up to this rant, in which she goes into further depth about the Fandom Jerk technique of accusing someone of being "racist against sidekicks."