Fortunately the wait and see method is successful most of the time with birth palsies. However, too much optimism can lead to too much waiting and delay appropriate care. The timing and degree of muscle recovery in the first 3 to 9 months of life is very predictive of recovery and highlights those infants who may need nerve surgery. If your child has a Horner's syndrome (different size eye pupils, eyelid opening), then the prognosis is poor and you need to see a specialist in the first 3 months of life. Otherwise, you should monitor your child's ability to open his or her fingers, lift his/her arm to the shoulder level, and raise his or her hand to their mouth. Also, be certain to keep the range of motion of all joints full in order to prevent contracture or shoulder dislocation. It is very helpful to work with a therapist with experience with infants with brachial plexus birth palsies. If there is uncertainty, please seek consultation with a specialist in this area.
(UBPN thanks Dr. Peter Waters for contributing this answer)