I understand your fear of spiders in your yard when a little one will be playing. We have a 3 1/2 year old, and last summer it felt like both our yard and our house was overrun with very large wolf spiders, the kind that build large webs in the trees that sometimes reach all the way to the ground. Well, this year I vowed not to let them freak me out, and I've done a lot of research on how to reduce their numbers without using pesticides. The most interesting and important thing I've learned is that spiders' taste buds are in their feet, and they will avoid areas like the plague that taste the least little bit of something they don't like.
Spiders had already started coming in our house just a few weeks ago when I decided to take action, and I sprayed all around our window and door frames, and along the baseboards and anywhere else I could think of with lemon Pledge, believe it or not. And I wasn't using the name brand stuff, either. Just the cheapie one, but I did make sure it had real lemon oil in it. And I kid you not, the spiders have disappeared from inside our house. I have also read that Dr. Bronner's peppermint and lavender soaps work great in a diluted solution of water, mixed with a few drops of essential oil (think strongly scented, like citrus, lavender, cinnamon, citronella, tea tree oil, etc.) I plan on using this on my son's playset outside, because it won't harm him to get it on his hands and eventually in his mouth.
I also sweep down the underside of my porch roof and underneath any overhangs about once a week to discourage them from forming webs. Also remember that birds, especially hummingbirds, eat spiders. Hummingbirds even use spider webs to help build their nests. I've put 2 hummingbird feeders and a regular birdseed feeder out hanging in the trees where I saw spiders nesting last year, and so far this year we have a lot of beautiful hummers, robins, and cardinals in our yard, but very few spiders.
There's no way to get rid of all of them, and you wouldn't really want to, because they are such beneficial insects, even though they are kind of creepy. But I think you can keep them at bay and out of your daughter's playset relatively easy with a little maintenance.
I hope this helped!