Granted I don't mind kids in my home salon, and often then not work with my own kids in the next room (it is their playroom), I think I've been very lucky that clients always ask if it's ok if they bring their kids with them. All moms always make sure their kids stay in the playroom next door and don't go upstairs or run around my work room. My kids love company and it works out great actually. That said I really do have lovely clients, and their kids seem to be very well behaved. Not everyone is like that unfortunately. One lady once brought her very loud 3 and 5 year old kids to the salon where I work for a FACIAL! All I can say it was a nightmare for everyone. The kids were running all over the salon, screaming, touching and grabbing on everything, with all the staff and even other clients trying to quite them down, while mom couldn't even get up from the bed. Total disaster! That day I thought that some policy regarding kids would've been great, lol.
So while I'm the mom that usually has to bring the younger one with me everywhere, I always make sure it is ok with the place where I'm going. That's why I went to my hairstylist at home, where she welcomed kids, for a couple of years, and not went to a hair salon. So for me a gentle notification that kids are not allowed would've been enough to either choose a different place or to arrange a time when I can be by myself.