LF – there are clearly a number of respected Poskim who allow shaking hands in a business setting, as well as a number who do not. If one is asked why they do/don’t shake hands when Moshe Cohen, an Orthodox Jew, does the opposite, the best thing to say is that there are different authorities and views, and he follows a different view. It’s not a big deal, and I’ve never had an issue using this reason in a number of contexts (including explaining why Chasidim don’t shave, while other Orthodox Jews do).
As to what to say when not shaking hands – one thing I heard many years ago is that one who does not shake a woman’s hand should say “I’m sorry, but for religious reasons, my wife doesn’t shake hands with men, and I don’t shake hands with women”. By phrasing it that way, you hopefully avoid insult by making it clear that the issue is not that you consider women unclean or beneath you – the restriction cuts both ways.
an Israeli Yid