This is just some simple research to say yes its possible BUT I have to say I own 2 dogs and a cat and I support animal ownership I do not post this to be against animals. I think our decisions with anything in a life with CF all depends on how much of a bubble you want to live in, I choose not to live in a bubble.
On to what I found, I narrowed down the article to make a point, feel free to read the full article at the link below if you're interested.
"In 2003, a 31-year-old female nurse who had psoriasis was identified as an MRSA carrier during the above-mentioned MRSA outbreak..... Screening her home environment showed that her 1-year-old daughter, who also had psoriasis, was colonized in the nose, throat, and skin lesions. The nares, but not the perineum, of their healthy pet dog were also colonized...... The staphylococci isolated from the nurse, her child, and the dog were all identified as
S. aureus...."
http://wwwnc.cdc.gov/eid/article/10/12/04-0387_article
So my point being carriers are everywhere, so if you have one: love your pets, love your spouse, love your kids. Don't isolate yourself so much that you stop living.
I agree with sroeseler, if you are concerned with a certain breed due to known skin issues, research around and pick a dog that fits your(our) situation.