Why do we swing our arms when we walk? Like the others have said, it's probably to keep balance. Try running, and you'll be swinging your arms like crazy. If you try to run without swinging your arms, I believe you are more likely to topple over since you're not balanced. Notice that when you step with your left foot, your right arm tends to swing up and when you step with your right foot, your left arm swings up. That convinces me that you do it for balance.
Also, you made a point about babies being quadrupedal before being bipedal. If you observe a baby, as well as any other quadrupedal animal, its right forelimb and left hindlimb are synchronized in motion, just like its left forelimb and right hindlimb.