I'm not sure what your argument is. Unless you are part of HR in your made-up hypothetical situation, it's not your job to intervene or seek retribution because you'll just screw it up and you'll make things worse. Personally, if you know both parties well you may very well find yourself skeptical of those claims ... or not - context matters.
All companies will have a process for handling harrassment claims and process for resolving them. Accusations should be directed to HR (or suitable designated party at the company). If the harrassment continues you may have to involve the legal system - that's what it's there for.
So, your answer in this case is: you do not get involved, period. Deflect, maybe some logistical support to direct them to HR, but nothing more?
They go to HR and a week later they're let go – no real explanation given. (Hey, it's at will employment here in CA.) Or, as happens at Uber, nothing happens, and your coworker continues to complain about harrassing behavior (if they're even talking to you at this point). What do you do now?
My point is: the real world is messy, and doesn't come with cut-and-dried definitions and solutions. Processes can help some, but true neutrality comes at a cost. Corrupt and sexist agencies in your company can continue to operate with aplomb. Your relationship will be damaged with coworkers who have had this happen to them – to say nothing of how it damages them when they have this happen and all of their coworkers look nervously away. It can poison your relationship to the company and all the good people that work there.
I read "please believe women" as an invitation to get involved in a constructive way, and I fervently disagree with you that you as a coworker can only make things worse. You can offer emotional support, so your coworker knows you're there for them. If you have pertinent information, you can offer it. You can hold your company's feet to the fire to make sure they're handling it appropriately. And of course – since I didn't specify the gender in the scenario – you can do the same whether the people involved are male, female, or anything in between.
All companies will have a process for handling harrassment claims and process for resolving them. Accusations should be directed to HR (or suitable designated party at the company). If the harrassment continues you may have to involve the legal system - that's what it's there for.