I had a different post planned. It was originally going to be just "age 7, not any older" - short & sweet; but after reading the other moms' posts I have a different opinion about how old is too old - I've raised that age from 7 to now 10 and will continue to base it on a place-by-place basis. However, I did ask my daughter her opinion just to get a kid's thoughts and it is below.
My daughter is 9.5 and my son just turned 6. I know this was about boys, but it applies to girls out with Dad, as well. I started to let each of them go on their own at about 5.5 years. Our daughter stopped going into the mens' room with my husband and started going into the womens' alone when she was about 5.5 and my husband would wait outside. My husband said the men would notice her come in and literally high-tail it out of there. It should be the same with the boys. Someone here is going to get upset that she went in there, but where was she supposed to go when I wasn't with them? They hadn't started making Family Restrooms available, yet, and women would have made a fuss if he had chaperoned her in the womens' room. Same question as when Dad isn't with our son.
With my son I usually let him go alone if it is not very busy (WalMart, HEB or a restaraunt where we can monitor who goes in and out). For the most part, though, I usually take him with me, but he's getting modest already. I'll probably quit taking him completely about age 7 (I've changed my mind to 9 or 10 after reading other posts). I don't cover his eyes in the womens' room since there are doors on the stalls and I've taught both of my kids it is rude to try to peek. I don't normally see women running around half-dressed outside of the stalls.
I asked my daughter, who is 9.5, this morning how she feels on this topic and should her Dad make her use the mens' room if I'm not there. She looked at me like I had two heads. She would be totally embarassed - her words. She was horrified at the thought of running into one of her male classmates (aged 9-10, 3rd grade) in the womens' room or if she ran into one while going into the mens' room. She said she would rather try to hold it until she got home. We need to think of our kids' and other kids' emotions, too, not just the fears of grown-ups.
I'm going to be very unpopular with this, but I do have a problem with the 10-year-olds and older coming in just based on how the kids feel about it and that they are hitting puberty. Would you insist that your 9- or 10-year-old daughter go into the mens' room with Dad or Grandpa since there are sick women out there, too? Yes, there are women who hurt kids, too. It IS the same question because my husband spends what we call Daddy-time-with-the-kids/Mommy-time-without-the-kids. I am not embarassed for myself that the boys are in there, I am embarassed FOR them because they feel humiliated about being treated like a baby.
As long as you can time and monitor them from right outside the door and hear them yell your code word, then you should be able to let them go in on their own, just like Dad has to do. You CAN go in if you feel the need. I know it's a big bad world out there, but we do need to teach them to do things on their own and how to watch out for themselves.