Is it permissible according to Halacha for a man to shake a woman's hand?
The Ben Ish Hai (Rabbi Yosef Haim of Baghdad, 1833-1909) addresses this question in his work Od Yosef Hai (Parashat Shofetim, 22; listen to audio for precise citation). He records there the practice that had become customary in Europe for guests to tightly grasp the hands of the host and hostess upon arriving in the home. The Ben Ish Hai writes that since a handshake serves to express mutual feelings of friendship and affection, it falls under the Halachic category of "Derech Hiba" – affectionate contact with a member of the opposite gender – and is forbidden.
In this context the Ben Ish Hai cites a comment of Rabbi Yehuda Ha'Hasid (Germany, late 12th-early 13th century), in his Sefer Hasidim (1090), forbidding shaking hands with a gentile woman, even if the woman's hand is covered with a glove. The Ben Ish Hai explained that although the Sefer Hasidim speaks here specifically of gentile women, it did not intend to permit shaking hands with Jewish women. Rather, the Sefer Hasidim seeks to dispel the possible misconception that one would be allowed to shake a gentile woman's hand in order to avoid the ridicule that might result from refusing to shake her hand. The Sefer Hasidim thus emphasized that a man may not shake hands even with a gentile, but it goes without saying that shaking hands with a Jewish woman is forbidden.
Thus, it is strictly forbidden for a man to shake a woman's hand, regardless of whether or not she is Jewish; this prohibition applies even in professional contexts, where the accepted protocol is to shake hands even with members of the opposite gender.
