Frankly, it is not clear to me why you are so concerned about being falsely accused of sexual harassment. Yes, there is a chance of this happening, but the chance is very small, even compared to other equally or more grave things that are largely out of your control (serious health problems, accidents, and so forth).

Lately, maybe because of the news, I am afraid that if I displease one of the students (for example fail her) she might accuse me of harassment to hurt me or force the University into a deal.

How often does a student fabricate a sexual harassment claim against a faculty member who has behaved entirely professionally? Such a student has a lot to lose as well.

No matter how it ends up, it will have serious consequences on my life (personal and professional).

I don't really agree. Such allegations are treated confidentially at first, and if they are totally without merit they need not go public. I speak here from direct experience with the process for dealing with harassment allegations at my university. I have reported (as required) secondhand information about possible harassment by faculty members at my university, and these investigations were indeed kept confidential and the faculty members remain in good standing at the university.

It happened to a colleague (different university) and it practically destroyed his career and marriage; even though he was finally cleared.

I'm sorry to hear that; maybe that's what's setting you off. I don't know your colleague's situation at all, so I can't speak to it. I do however respectfully disagree with your claim that any false claim of harassment by a student will come close to destroying one's personal life. There are many public examples of marriages that survived sexual harassment. Moreover, there are people in my life whom I would not believe guilty of sexual misconduct based on any amount of circumstantial evidence. I think a lot of people feel this way about their spouses.

I thought of rejecting any female students (you can select the students to accept), but I could never discriminate like that. I thought of hiding a camera in my office, but that could go badly if it's detected (and it's illegal). Asking only the female students to meet me at the library or public space would seem very odd and discriminating. Asking all students to do the same would be difficult to me.

I agree that with the possible exception of the last, these are terrible ideas. (Really, you thought of putting a hidden camera in your office to protect yourself from claims of sexual harassment?) The last idea is not inherently bad, but...how is it different to have students meet in a library than to meet in your office with the door kept open?

How do you protect yourself from this? How can I not discriminate, be a good educator, but protect myself and my family? Am I being paranoid (probably)? I am uncomfortable asking senior academics in my department about this.

"Paranoid" is a little strong, but I don't really understand where your worries are coming from. Nor do I understand why you are reluctant to talk to your colleagues about this. You seem to be slightly "hung up" on something here.

Anyway, here is my strategy for avoiding allegations of sexual harassment. The first two points are key, and the third is optional but helpful. Here we go:

1) Never sexually harass your students. Don't even come close -- have a clear-eyed view of what the boundary of acceptable behavior would be and make sure that you stay two steps away from that boundary. Especially, maintain a very strong sense of what is acceptable physical contact with a student. (Handshaking: okay. Tapping someone on their clothed arm or shoulder to get their attention: probably okay, but monitor while you do it to make sure that it is being received that way. Almost anything else: as a rule, don't do it.)

2) Make sure that your interaction with students looks to all observers like you are a steadfast subscriber to point 1) above. So, yes: keep your office door open whenever students are in your office. Feel free to make a point that you are doing so. Make a point of not talking about romantic relationships with your students -- either theirs or yours -- except possibly in ways that are so passing and innocent that they actually reinforce that you know where the boundaries lie.

3) Actually be ahead of the male-academic curve when it comes to knowledge and sensitivity about such issues.

In your case, I caught a few minor things that suggest that you have some room for improvement on the latter point.

Most of all, being fearful of being falsely accused of sexual harassment suggests a certain lack of empathy with your female students. I am aware that pointing that out is not directly helpful, but I do hope you can attain a better state in the fullness of time.

For every female student who falsely accuses their male faculty member of sexual harassment, how many female students are truly sexually harassed, or borderline sexually harassed, or not harassed but treated differently from the male students in a way that makes them feel uncomfortable? In my experience, if you include all of the above the ratio is something like 1:100 or more. If you can come off as legitimately sympathetic to these issues, then (apart from other benefits!) you make yourself a much less appealing target for students to make up stories. And having other students step in and say "On the contrary, Professor A is one of the good ones..." will certainly help to defuse things if it comes to that.

"Some of the students are girls." Do you mean that they are under 18? If you do mean that, say that. It is no longer considered appropriate to refer to adult women as "girls." Probably you would not refer to your male students as "boys," and by the way, you shouldn't.

"I understand that the same accusation could come from a male student. My 80s small town bias doesn't allow me to consider that possibility." What the what?!? First of all your statement is literally contradictory: you are evidently considering the possibility. Second of all: what are you trying to say -- that you're more backward / less progressive than other people in your position might be? You are giving an excuse that excuses nothing and that could be quite off-putting to many other academics. So don't say things like that.

Added: Here are some statistics on the prevalence of sexual harassment in American universities. I can't find statistics on prevalence of false accusations of sexual harassment against American faculty members (and doubt such exist), but in my best judgment it somewhere between two and three orders of magnitude more likely that a female student gets sexually harassed by someone in the university than makes up a claim of being sexually harassed by a faculty member. Several people have suggested that such statistics are not relevant to the OP. I respectfully disagree: as others have said, a rational approach to this problem is to quantify the risk and ask what tradeoffs the OP is willing to incur to lessen it. Knowing that female students and faculty routinely occur a risk hundreds of times greater is something to take into account when making these considerations.