I'm guessing you don't have kids?..
Not only did I have/raise three kids, but I currently have 6 grandchildren. I have no problem with them using a public restroom that may have a gay person in there or a transgendered person. First off, people are only in there to take care of nature's calling, wash their hands, maybe their face, and in the ladies room, dink around with their make-up. If people use a stall what's the issue? I don't think anyone in there is really checking out people that closely and if you are, that in itself is kinda weird.
People have the right to simple human dignities. And just because someone is born who is different from you, it doesn't make it alright to segregate them, or punish them or deprive them.
If you look at the stats of those who molest children, your real concern should be with straight males and other family members.
And if you would care to perhaps read a few blogs, look at some scientific data, and look at some websites/organizations to educate yourself a bit, you would learn that trans people don't want to stand out, they mostly just want to lead a normal life like everybody else and fit in with others. If you think that being gay or lesbian is tough for kids and young adults, you can only imagine that it is ten time more difficult for transgendered human beings. They've been hiding, afraid, threatened and shunned by their own families, classmates, and more. Most of them have huge issues just trying to get a decent job.
No one should care or judge anyone by their gender or their sexuality. Judge people on their honesty, are they trustworthy, do they have a good heart, are they decent, do they have ethics and courage, in other words are they a good person. Besides, the numbers of transgendered people is so small, that you have a better chance of being struck by lightning than sharing a bathroom with them.
Our kids and grandkids need to be taught about the world in stages, appropriate to their age and maturity level. They need honesty and truth. They need to learn acceptance. They need to know that everyone is different in many ways, but in the important ways of how we are human and how we connect as humans, we are all the same. We all have the same feelings, hopes, desires, fears and anxiety. We ALL share all of the emotions that make up a human being. And we all have the ability to love. We all want love and acceptance. With such a strong connection, it should be no big deal what gender a person is or whom they choose to sleep with.
If you read up on, learned something about transgendered men and women, maybe even met one and talked to them, you'd surely see that there is nothing wrong, oddball, strange or scary about them. You seriously have nothing to fear or be concerned about.
Let me enlighten you on something. Transgendered people are murdered in this country on a regular basis. They are also denied fair housing in renting a place to live, they are denied jobs, they are denied service in certain restaurants or bars, they are bullied, they are denied many things. The last damn thing they need is to be denied a GD place to take a leak or a dump.