The line I’ve used is “for religious reasons, my wife doesn’t shake hands with men, and I don’t shake hands with women”. By first mentioning that my wife doesn’t shake hands with men, I’ve avoided making them think that I think that women are lesser than I am.
FWIW, I can also say that I was told by a well-known Rosh Yeshiva, widely accepted as one of the premier Poskim in the American Yeshivish world, that if it’s necessary, I can shake hands with a woman in a business context, since it’s not Derech Chiba. Accordingly, when necessary or to avoid causing offense (such as when traveling in Europe, where religious restrictions are not as widely accepted), I’ll shake a woman’s extended hand – though I’ll not extend mine first.
Good luck to you,
an Israeli Yid