So why are hospitals bothering with RFID bracelets? I think they're primarily to reassure the mothers. Many times during my friends' stay at the hospital the doctors had to take the baby away for this or that test. Millions of years of evolution have forged a strong bond between new parents and new baby; the RFID bracelets are a low-cost way to ensure that the parents are more relaxed when their baby was out of their sight.
Security is both a reality and a feeling. The reality of security is mathematical, based on the probability of different risks and the effectiveness of different countermeasures. We know the infant abduction rates and how well the bracelets reduce those rates. We also know the cost of the bracelets, and can thus calculate whether they're a cost-effective security measure or not. But security is also a feeling, based on individual psychological reactions to both the risks and the countermeasures. And the two things are different: You can be secure even though you don't feel secure, and you can feel secure even though you're not really secure.
The RFID bracelets are what I've come to call security theater: security primarily designed to make you feel more secure. I've regularly maligned security theater as a waste, but it's not always, and not entirely, so.
I'm not saying that security theater isn't a lot of the reason for the RFID bracelets, but you do need some kind of tag for people's babies, which, after all, look pretty much alike. Tags let you ensure that you match the right mother with the right baby. Obviously, you could (and for many years people did) get away with old-style non-RFID plastic bracelets, but it's probably not that much more expensive to make them RFID, especially since it saves you the trouble of having more expensive security theater—guards at every exit. And of course having RFID tracking means that you can use the system to track where infants are even inside the hospital, which presumably is useful if/when you lose track of patients.
The leading systems seem to be made by VeriChip. They make infant protection gizmos, Hugs which had a cut-detecting band which triggers an alarm if it's tampered with and HALO which works with commodity bands but senses if it's attacked to skin. A related product is RoamAlert, a "wander prevention" solution designed to let you keep track of patients in nursing homes (and presumably mental hospitals).