From Torah.org:
Question: Is it ever permitted to shake hands with a woman? Is there a dispensation to do so if otherwise one would suffer a substantial loss or would embarrass the woman, possibly causing a chillul Hashem?
There are, however, some situations where a handshake is offered as a matter of protocol, such as an introduction to a customer or an employer, to a doctor or to a distinguished politician. In these situations, the handshake is not a sign of affection, friendship or a personal relationship and would, theoretically, be permitted. Still, the poskim are in agreement that one must do whatever he can to avoid shaking hands under these circumstances as well. This is because the yetzer hara for arayos is overwhelming. An innocent handshake may lead to a casual embrace; a harmless introduction may blossom into a full-blown illicit relationship. It is extremely difficult to define what is and what is not derech chibah when it comes to a handshake, and it is, therefore, the consensus of the poskim to be stringent in this matter{6}.
Under extenuating circumstances, e.g., one would lose his job were he not to shake hands with a female customer, or if, by refusing an extended hand, one would publicly humiliate a prominent personality, there are some poskim who find some room for leniency to return a handshake, if the hand is proffered in a manner which is clearly not affectionate. All poskim agree that one must do whatever he can to avoid being caught in such a situation.
All of these halachos apply equally to men and women.