The Shulchan Aruch clearly states that breastfeeding women should fast on Yom Kippur in the same manner as others (Orach Chaim 617:1). As in all areas of halacha, if there is a medical risk, there is room for leniency in individual cases. A rabbi must be consulted in such situations.
Fasting by a breastfeeding woman raises questions as to the health of two people, the mother and the baby. Most breastfeeding women can fast without any risk to themselves. Many report, however, that fasting while breastfeeding is harder than it is at other times. They should make all efforts to drink a lot the day before the fast and to rest as much as possible during the fast. It is more important to fast than to attend shul. Couples with young children that need to be cared for during the fast should think about how to do this without overtaxing the breastfeeding fasting mother – hiring a babysitter for example, or even having the husband remain home for part of the day to help. Women should not reduce the frequency of breastfeeding during the fast, as this is likely to lead to a breast infection.
As far as the effect on the nursing infant, there is almost no data on the effect of such a fast on human breastmilk. The only published article to date is by O. Shefi: "Tzom Yom Hakippurim v'Hashpa'ato al Nashim Menikot," in Assia 14, Elul 5754. In her review article, the author found one human study that showed a slight difference in milk composition after an overnight fast. Studies in animals showed changes in composition and a small reversible decrease in milk supply the day following a 24-hour fast. The author reported anecdotal data from women she interviewed of reduced milk on the day after the fast. However, no statistical data are provided. Based on my own experience, as a pediatrician, lactation consultant, and breastfeeding mother, I feel that most babies can weather the fast with little more than slight fussiness. Mothers can help by being prepared with a bottle of expressed breastmilk from the day before that can be administered at the end of the day (either by bottle, syringe, cup, or spoon, depending on the age and nursing history of the baby). Mothers should arrange to have a quiet day the day after the fast when they can spend time with the baby and nurse more frequently to make up for any slight decrease in supply that might have resulted from fasting.
These are general guidelines. There are individual cases such as sick infants, difficult nursing situations, and past bad experiences where mothers should be told to drink in shiurim (small quantities). Any such case requires an individual medical assessment of the situation and an individual halachic ruling.