An LLC does not reduce your liability if a baby gets hurt. All it does is protect you from the business's financial liabilities, and in most cases also protects the business from yours. For example, if some parents didn't like your work and won a lawsuit against the business, damages would effectively be limited to the business's assets.
Oh the other hand, torts are always personal. To protect yourself and your business from torts like injuries and other non-financial liabilities you need to buy insurance. Especially if you're operating a business like that out of your home, you should look into professional liability insurance on the business side and umbrella insurance for your family. Talk to a broker about how much you need and get some quotes.
Also, you should think about talking to an attorney about getting a standard liability release form drawn up. This doesn't necessarily prevent someone from suing you and winning, but it's another layer of protection.