There are allergy meds for kids to help with pet allergies, as well as outdoor allergies and air pollutants, but I understand not wanting to have your child on them. I can tell you that my mother is allergic to pet dander (as well as having seasonal allergies) but we've had a cat as long as I've been alive. Only after I had braces at (seventeen thru twenty) and developed an allergy to latex due to exposure did I develop allergies to pet dander and it's only when
they rub on my face. I can pet them and touch my face without issues. (Note: I had seven cats when I developed the allergy)
Like Dirge said, bathing the cat helps absolutely. Vacuuming the carpet and furniture regularly is another really good way to cut down on dander if allergies are a problem. I know it's super expensive, but the
Rainbow is the best machine on the market if it comes to that. It uses water as a filter in stead of traditional vacuum filters so almost nothing gets pumped back into the air, like what can happen with an old traditional vacuum. It also acts as an air purifier. You can schedule someone to come to your home to demonstrate the machine before you buy it. Speaking of... using air purifiers is a good idea too.
Finally, if no one in your family has a history of allergies and you only have one cat, I doubt it will be a problem.