Given the comments which include lots of different aspects (some very different scenarios and motivations were addressed), another — sorry, rather long — take on the question.
What to do when you see sexism everywhere and want to do something against it?
There is a basic psychological process called confirmation bias. People usually look for information confirming their beliefs. This heuristic makes sense in everyday life, but is bad for getting an objective picture of a given situation. Regarding the current topic, it's actually very easy to see sexism everywhere — the hammer-and-nail thing. We have pretty much built science to combat that tendency.
So if you work in Academia, do better. Act like an empirical scientist. Do not let fears/wishes/expectations determine your perception, but gather actual data. Try to be objective and protect yourself against your confirmation bias. This means that the stronger you believe something, the more strongly you have to be wary of being influenced by your expectations.
For example, when it comes to interruptions, count the number of times it happens, and also count the number of times women interrupt (no matter how well it was 'justified') and how often (usually low-status) men are interrupted. I am pretty sure it's a status thing, not a sex thing.
Same with criticism. There is the soft bigotry of low expectations and there is benevolent sexism in the sense that women "have to be protected" — which might lead some people to see criticism as more harsh when it is directed at women. However, harsh criticism is just normal in science. And yeah, sometimes unfortunately. But it is also very much needed. We need bullshit detectors — and the ability of science to cull bad ideas is what makes science great. Feedback that addresses the quality of the work is great — no matter the motivation. It's the only way to improve.
So, when it comes to instances of perceived sexism, an empirical scientific mindset is helpful: Question, gather data, then judge. Social interactions are fraught with misunderstandings. Yes, sometimes one person is at fault, but usually both contribute to the situation. In general, use the principle of charity. Assume an honest misunderstanding (or fuck up, humans are not machines, and even those have glitches) — not evil intent.
Also, with few exceptions (which are cases for the police), "The person felt it so it must be true" is very bad reasoning. I've encountered too many people — men and women — who take any criticism of their ideas as an attack on themselves as a human being. That's a bad way to live and makes it impossible to do science.
And above all, deal with potential situations individually, not collectively. Treat people as individuals, not as members of groups. If you really think specific groups (like old white male academics) are a problem think about what this makes you. Don't use individual cases to vent your frustration about perceived general injustice. Displaced aggression helps no one.
Also don't focus on exaggerated cases or cases where you'd need a magnifying glass to see the sexism. That hurts the fight against actual sexists. And let's face it, there are some — both men and women — who are sexist against men or women. And yeah, in science, the quality of the work, the strength of the evidence and arguments should matter, not the sex of the person who does science.
And although it's hard, don't take the cases personally. Try to maintain some distance. Humor is great for this. There is a reason for gallows humor.
And above all, don't instrumentalize other people (usually women) as victims you have to fight for in order to feel moral. Cases of sexism are not there to make you feel morally good.
If sexism happens to you, address it with the person who acts sexist. Again, question, don't immediately assume the worst. If you always assume the worst, you will confirm your belief, and while you might feel righteous, you will never be happy. The world we need is one in which when one person fucks up, others give feedback and the issue is resolved without a fuss. And a competent human being has to deal with problems in human interaction.
If you perceive a person encountering sexism, then stop. Is the situation really as you perceive it? Do you really have a clear grasp what people actually feel (quite a degree of empathy needed) and think (hello mind-reading)? Do you really have all the facts (e.g., other factors that might influence a course of events)? Consider that post hoc ergo propter hoc is a fallacy, no matter how true it feels. There might be other reasons for a strange turn of events. So by all means, question. A university where you cannot question is a strange place.
And if it is really sexism, does the "victim" really not have the ability to deal with the situation on his/her own? Don't remove the perceived victims agency — their ability to learn to deal with these situations or demonstrate their ability to do so. It's very condescending and disempowering. Consider that helping in the short run might impede the person's ability to deal with other situations in the long run. Almost all people want to do good. But wanting to do good and doing good aren't necessarily the same.
Again, we should strive for competent individuals. Not only when it comes to sexism, but also when it comes to speaking truth to power. After all, a university is the one place that looks for truth (without ever finding it). It requires a professional environment in which merit counts and all other aspects (e.g., sex, race, etc.) are neither a positive nor a negative. And to actually go looking for the truth without being influenced by political or social opinion does require strength. And you don't gain strength by being protected all the time.
