Epidural anesthesia does affect the baby.  Any anesthesia that is administered to the mother reaches the baby and does have consequences.

Studies show that epidurals cause abnormal fetal heart rate, sometimes severe, which may occur in association with or independent of maternal blood pressure.  Incidence of abnormal fetal heart rate with use of different drugs is:

43% - bupivacaine
16% - chloroprocaine
10% - lidocaine

Because epidural anesthesia alters red blood cell properties, babies sometimes exhibit jaundice as a result.  Also, babies of mothers who had induced labors as well as epidural anesthesia had even poorer scores.

Epidural anesthesia also increases the rate of labor complications and interventions.

During the first day after delivery, medicated babies perform poorly compared with control group babies.  Babies belonging to mothers whose labors were induced or where forceps delivery was used showed even more depressed scores.  By the fifth day after birth, scores had improved, but medicated babies still showed poor state organization (crying, feeding, alertness, sleeping).

One month after birth, the mothers of these babies perceived them as being more difficult to care for.

Sources:

Goer, Henci.  (1995).  Obstetric Myths Versus Research Realities:  A guide to the medical literature.  Westport, CT:  Bergin & Garvey.

Abboud, T.K. et al. (1984).  Continuous infusion epidural analgesia in parturients receiving bupivacaine, chloroprocaine, or lidocaine - maternal, fetal, and neonatal effects.  Anesth Analg, 63:  421-428.

Clark, D.A. and Landaw, S.A.  (1985).  Bupivacaine alters red blood cell properties:  a possible explanation for neonatal jaundice associated with maternal anesthesia.  Pediatr Res, 19(4):341-343.

Murray, A.D. et al.  (1981).  Effects of epidural anesthesia on newborns and their mothers.  Child Development, 52:71-82.
 

Epidural anestheia CAN in fact cause blood pressure to drop during labor, can slow down contractions making labor augmentation necessary, and is in fact ineffective for some women.  Either they receive no pain relief, or only certain spots are numbed.  Other frequent effects are spinal headache and backache which persist weeks or months after the birth, and in some cases never go away.