The Shulhan Aruch (Yoreh De’a 265) cites the practice of the Geonim to recite a Beracha over a cup of wine at a Berit Mila. This Halacha does not appear in the Talmud, but it has been accepted and codified by the Shulhan Aruch, and this is, indeed, the custom. After the circumcision, somebody – usually the Rabbi – recites the Beracha of "Boreh Peri Ha’gefen" over a cup of wine, followed by a Beracha over Besamim (spices) and the Beracha of "Asher Kidash Yedid Mi’beten." After these Berachot, the baby is named.
Unfortunately, it sometimes happens that as a result of the various blessings that are recited, nobody drinks any of the wine from the cup. If nobody drinks from the cup, the Beracha of "Boreh Peri Ha’gefen" is considered to have been recited in vain. And although it is customary to give a drop of wine to the baby, this does not suffice, since the baby is quite obviously not obligated in the recitation of Berachot. The cup must be drunk either by the person who recited the Beracha or by an adult who heard the Beracha and did not make any interruption until drinking. This means that if somebody else drinks, he must ensure not to recite any of the Pesukim that are customarily recited by the attendees during the recitation of Berachot ("Hodu Le’Hashem Ki Tob," etc.). In some places it is customary for the baby’s mother to drink from the cup of wine, especially when the Berit takes place on a fast day, when only the mother is permitted to eat and drink. As mentioned, if the mother will be drinking but the person who recited the Beracha will not be drinking, the mother must have heard the recitation of the Beracha and not have spoken at all in the interim.
It should be noted that there is no minimum amount of wine that must be drunk from the cup at a Berit Mila; it suffices even to drink a small sip.
These Halachot are discussed in Orech Yamim, p. 142 (listen to audio recording for precise citation).
Summary: It is customary to recite Berachot over a cup of wine at a Berit. If the person who recited the Berachot will not be drinking, then another adult present must drink. That person must hear the Beracha over the wine and ensure not to speak at all until drinking. It suffices to drink even a small sip of wine from the cup.
