So you made a mistake

So, you made a mistake. You gave a porny presentation at a technology conference, or you made a sexist blog post, or you organized a conference with 0.00% female speakers. And, you got called out on it. What can you do?

Don't dig any deeper

• You may feel unfairly singled out or that critics are overreacting. There's a temptation to be dismissive or defensive; avoid that temptation.
• Take some time to think about the responses and make some conclusions. Speak in haste, regret in leisure, as they say. If you feel the need to address a controversy immediately, a simple, "I'm reviewing comments now and will have a statement in a few days" can be fine.
• Ask your friends and allies privately not to fuel the flames: they will naturally be inclined to vigorously defend you, possibly stating positions or attributing motives to you in ways that you will not like or that will make everything worse.
• Respectfully ask the opinions of a few people you trust. Try not to bias this group towards people who you think will agree that you did the right thing. Note that female friends, in particular, will be put in a hard place, as they don't necessarily know that you're inclined to take their opinions on feminism seriously. It may help if you acknowledge that you're inclined to take the criticism seriously. Don't badger someone to help you out.
• Do 101-level research yourself if you can. For example, instead of asking a woman you know "Is calling women 'babes' really sexist?" try googling that same question first, reading and considering some of the top results, and seeing if that answers some of your questions. Doing some of the work first may make some of your trusted friends more willing to be open and helpful.
• Take care of yourself. If you're emotionally upset, turn to the relationships you would normally turn to for emotional support. The impact of a lot of unsolicited criticism can be overwhelming, and some of the same self-care steps recommended for the targets of internet trollstorms can apply. Making a public mistake does not necessarily make you a terrible person, and you're not required to feel terrible about yourself while you're fixing your mistakes.

Take responsibility

• Apologize sincerely
• Don't ask that all the wrongs done to you by the criticism be redressed. Say, for example, that some of the criticism attacked you personally and painted you as a bad person rather than criticising your specific action. This really really hurts, but adding riders to your apology about "hurtful personal criticism" or "I agree that this action was sexist, but that doesn't make me a sexist bastard like so-and-so said in her blog" will detract from your apology. If in your considered opinion some of the criticism was too strong, was badly phrased or was outright cruel, just leave it alone and express your contrition about the parts of the criticism you agree with.
• Don't encourage further damage. If you were selling a racist shirt, stop selling the racist shirt. If you posted an inaccurate news story, post a notice of the correction on the story, and a link to the apology. If it's impossible to edit the original (such as a tweet), consider whether it is less damaging to people who are affected to take down the original or post some form of reply. If it's a thing that you did in person, don't do the thing again until you have considered your actions and sought advice.
• Resist the urge to try and purge the internet of incriminating material (presentation slides or video, blog screencaps, etc), especially material outside of your control. What's done is done; trying to get things taken down makes it look as though you are attempting to evade responsibility. Besides, the internet is great at remembering things in perpetuity. Better to spend your energy on learning from the incident and making genuine changes that will in time distance you from it.
• Accept that an apology, no matter how sincere and how thoughtful, may not solve everything or even anything, that you may have permanently alienated some of your readers/users/colleagues.

Take pro-active steps

• If you can, at the time of your apology offer some guarantees about steps you will be taking to improve things in future. For example, "next year we have committed to a program committee with 30% or more women members," "we have engaged a trainer for diversity training at our organisation," "we pledge in future to ask speakers at our conference to adhere to a code of conduct," "some people I trust are going to have a look over my blog posts for the next couple of weeks and warn me if I'm being a jerk while I continue to educate myself." However, don't promise things you haven't committed to following through on, or ask other people to help out with your problems (eg don't say that you'll review slide decks if you haven't got a volunteer ready to review slide decks).
• Over some appropriate period of time, for example the lead up to next year's conference, announce what happened: demonstrate how you took those steps you promised.