We have a 3 yo. female staffordshire/american pit bull mix. When I went into labor at home, she refused to leave my side. She would follow me everywhere..it was almost like she *knew* what was going on. After I had the baby in the hospital, dp brought the baby hat and a blanket home for her to smell a day before I went back home. She smelled it and got very excited...running all over the apartment, etc. When i got home, all she wanted to do was smell the baby and smel me. The first few days she was very whiney and vocal..mostly because we were spending so much time and energy on the baby and not on her. After the first few weeks she had quieted down and her behaviors began to change completely. She has always been hyper and a bit of a spaz but around the baby she is incredibly gentle and sweet. She used to consider dp the "alpha" dog of the pack but after the baby it seems that I have now taken on that role. She loves the baby and will stay with him at all times. They really have become the best of friends, I've never felt uncomfortable with her and the baby together. She's my nanny dog! (they are always supervised of course)
The most important thing is to make sure you give your dog enough attention after the baby comes and to always keep an eye on them around the baby. It's hard to do in the beginning but it pays off in the end. At least it did with us

Good luck!