I think that in all but the most blatant cases it is very hard to do. But there are a few suggestions that might get you started.
First, note that some of the offenders don't leave tracks that can be followed. A person could, for example, be anti-Semitic and not advertise it, but just find subtle ways to disadvantage Jewish students. There is probably much more of that sort of under the radar sort of bigotry than the more open kind in academia, since it is generally sanctioned when found out.
But, students know, or suspect, at least, that they aren't being treated fairly, even if they have no effective way to complain about it or correct it. So, talking to other students in an informal setting is a good way to learn the scuttlebutt. "What do you think about Prof Buffy? Is he just goofy or a real problem?" If you have a specific concern talk to students who might share that concern.
But, there may also be some record of past misbehavior, either official or otherwise. Disciplinary action may be in the public record (or not). Even a web site like RateMyProfessor is a source of (not well vetted) information that might make you think again about an individual.
It is very difficult, of course, to expel bigots from the profession. Often the actions don't rise to the level that permits a tenured person to be fired. While the difficulty of forcing expulsion is intended, it has some bad side effects. The blatant cases, such as Robert Moore and William Shockley are well known and instructive. Brilliant in their fields, castigated for their social views, but never expelled from their professions. These are the easy cases, since they are very public. But most bigotry thrives in darkness.
I just came across a book on implicit bias that might give background: Uncovering Hidden Prejudice