As others have stated, unless your child has known allergies, you should be ok. However, speaking from personal experience, when you already have breathing difficulties, pet dander can contribute to those difficulties. Maybe you guys could have a dog that is "hypoallergenic." I have two miniature schnauzers, and they are perfect. They don't shed, are considered to be hypoallergenic, are very loyal, are very intelligent, and can easily fit in your lap.
The bottom line is: You want your child to grow up as "normal" as possible, and if having a dog is part of that for your family, make it happen. Dogs are fantastic therapeutic tools. I appreciate the fact that you are concerned about the impact an animal will have on your child in regards to his/her health. You're doing a great job!
Sincerely,
Regina Pruitt---37y/o w/CF; married with two biological sons; Delta F508; currently in the hospital for acute exacerbation/hemoptysis