I totally agree with what you are saying as far as trusting your employees, but your examples are a little different; a better example would be "If an employee of yours says that another employee stole their laptop, would you demand security camera footage before firing the other employee?"

I think the key thing to do, as a manager, is to start by beliving the employee making the complaint. Don't tell them things like "Oh, you must have misinterpreted" or "was it really that bad" or "I am sure he didn't mean it".

You tell them you take their complain very seriously, that you are so sorry this happened to them, and that you will take action immediately. Then, you investigate. Shockingly, lots of harassers won't outright deny it if you ask them about specific behavior (don't come out and ask them if they sexually harassed, but ask them if they, for example, said these specific words that they are accused of saying); they won't lie because they don't think what they did was actually harassment, they think they were just flirting or being fun. You then take action, whether you need to fire the person, reprimand them, transfer them, demote them, etc.