The Shulhan Aruch (Orah Haim 129; listen to audio recording for precise citation) rules that Birkat Kohanim, as a general rule, is recited during Shaharit and Musaf, and during Ne段la on Yom Kippur, but not during Minha. The reason, as the Shulhan Aruch explains, is that people normally eat meals in the afternoon that might include wine. It is thus likely that a Kohen will be intoxicated at the time of Minha, in which case he is forbidden from reciting Birkat Kohanim. The Sages therefore enacted that Birkat Kohanim should not be recited during Minha, to avoid the risk of a Kohen reciting the blessing in a state of intoxication.
In light of this reasoning, the question arises as to whether Birkat Kohanim should be recited during Minha on a fast day. After all, if the entire reason for omitting Birkat Kohanim during Minha is the possibility of intoxication, perhaps we should allow Birkat Kohanim during Minha on fast days, when Kohanim obviously are not drinking wine. The Shulhan Aruch writes, however, that the Sages forbade Birkat Kohanim during Minha even on fast days, in order to ensure that people would not mistakenly recite the blessing during Minha on other days. If Birkat Kohanim were recited at Minha on fast days, people might become confused and recite the blessing in Minha even on other days. To avoid such confusion, the Sages applied the rule concerning Birkat Kohanim even to fast days.
However, the Shulhan Aruch adds, this is true only when Minha is recited early in the afternoon on a fast day. When Minha is recited just prior to sundown, then it resembles the Ne段la prayer recited on Yom Kippur, when Birkat Kohanim is recited. People are not likely to mistakenly reach conclusions about the standard Minha prayer from Minha recited just before sundown, because Minha recited at this time bears resemblance to Ne段la. Therefore, on Tisha B但b, for example, or on Shiba Asar Be探amuz, if a congregation prays Minha close to sundown, in which case the prayer resembles Ne段la, Birkat Kohanim is recited. But on Yom Kippur, when Ne段la is recited just before sundown and Minha is recited earlier, Birkat Kohanim would not be recited at Minha.
This is, indeed, our practice. When a congregation prays Minha early in the afternoon on a fast day, Birkat Kohanim is not recited, but the blessing is recited when Minha is prayed within a half-hour of sundown.
Summary: Birkat Kohanim is not recited at Minha, except on fast days if Minha is prayed within a half-hour of sundown. A congregation that prays Minha earlier in the afternoon on a fast day does not recite Birkat Kohanim during Minha.
