So, perhaps you have decided to avoid shaking hands, at least to try to observe shomer negiah (avoiding casual touch with the opposite gender). What if someone of the opposite gender offers you his or her hand? What do you say?
This is one of the most challenging aspects of becoming observant and is often something people put off until they have already begun observing other mitzvos. However, men who wear yarmulkes might want to start observing shomer negiah right away to avoid giving the false impression that it is okay to shake hands with women.
I can say that it is probably easier for women to avoid shaking hands with men than for men to avoid shaking hands with women. I feel that it is empowering to avoid shaking hands with men, because one of the reasons for shomer negiah is to preserve a woman’s dignity. When in these situations, I simply say, “I’m sorry, but for religious reasons, I don’t shake hands with men.” I would include all of this information to avoid misunderstanding. I used to say, simply, “I don’t shake hands with men,” but then someone thought that I disliked men, since it wasn’t apparent at that time that I was religious. As the Baal Shem Tov said, “Words that come from the heart enter the heart” and are understood. A sincere concern for the person extending his hand and a wish to avoid embarrassing him will communicated by a sincere statement.
In the recent elections, a Chassidic man named Jason Bedrick gained a seat in the New Hampshire legislature. Amazingly, he managed to win the election even though at public events, he did not shake hands with women. When a women’s group wrote to him and complained, he sincerely told them that it was out of respect for women that he did not shake hands with the opposite gender. All the women in the club voted for him, and he won by these few crucial votes.