If a "hypothetical" man (physically) walked into the restroom behind your 8yo daughter....you'd be ok with that?
I wouldn't completely trust any adult stranger in a restroom with my hypothetical kid, regardless of gender or gender identity.
...but, if your sentiment is that a transgender person inherently presents some sort of greater risk to children than non-trans, you'd need to substantiate that with some sort of data.
Study finds no link between transgender rights law and bathroom crimes - The Boston Globe
...but as I noted before, this idea that I could simply wake up tomorrow morning and decide I wanted to identify as a woman, and stroll into the women's bathroom at the local Target and start taking pictures and assaulting people, and that those sorts of activities are now somehow being done with the states blessing due to laws that allow people to legally change their gender identity is a false notion.
As I laid out above, the process for doing so isn't simply stepping out on my porch, proclaiming "I'm a gal!" and then proceeding to hit the women's shower at the gym...that's a fallacy.
One has to find a licensed medical professional:
A) whose practice included counseling and treating gender identity issues
B) who must provide the name of their organization, state medical license #
C) who must sign off on it under penalty of perjury, and a $2500 fine and potential jail time
...so it's not like a person can complete the process in 5 minutes at an Urgi-care center or a quick visit to their GP.
Apart from the idea of a regular dude wanting to go through all of this just to get a peak at some naked women in the shower being a dubious proposition at best, I don't picture many licensed doctors being willing to gamble their entire lucrative career and professional reputation on putting their pen to paper on something like this unless they have a high degree of confidence in what they're signing off on.
Not to mention, many states have provisions that state that the legal gender designation change can't happen until the name has been legally changed (sometimes for a minimum amount of time) and that hormonal or surgical intervention has already taken place. (Like Missouri and New Mexico for instance)
So to recap:
The fear is that "Tom" is going to wake up and say
"Yknow, I'd like to go look at some naked women in the shower...so I'm going to go to a licensed therapist, change my name to "Tomina", start taking estrogen or get my body surgically altered, live that way for 90 days, so that I can get the proper paperwork to legally go into the women's shower starting February 1st"
Seems like quite a stretch...