Many Orthodox Jewish men will not touch women (outside of their spouse and immediate family).
I shake hands with women but I don’t enjoy it.
Around 2003 there was that "Ethicist" (Randy Cohen) article about that woman who wanted to void some business dealings with a frum Jew because he explained that he does not shake women’s hands. I remember Rav Schachter suggesting the "dead fish" appraoch when asked after shiur.
A few days later another member of the shiur told me that Rav Schachter paskined for him that it was assur to shake hands. It turned out that he was a member of the YU Bowl team, and customarily after games the two teams shake hands. In that case Rav Schachter felt that the hand shake wa in no way casual, and in fact when asked further he explained that the bachur represented YU, an orthodox institution, and it would send the wrong message if the bochur would shake hands with girls on the other team.
Also, Rav Schachter does state when certain rules fall under "common sense", and I don’t think that he felt this issue to be simply that. He doesn’t quote Teshuvos just for the sake of it.
Tzvee posts: This surely is common sense. There is no way that a handshake in a business situation, as gil has set it forth, has any romantic or sexual aspect. To interfere with normal transactions in a corporate office by interjecting an opinion here is a disruption to an individual’s ability to conduct normal commercial life, something that the halakhah does not have any business or interest interfering with.
Meir posts: Rav Schachter told me that Rav Soloveitchik shook women’s hands. He said nothing to me about the "dead fish" tactic.
Anon posts: I heard from the posek R. David Cohen of NY that if a woman sticks out her hand one should shake it. He rejected the possibility of trying to explain to her that you don’t do it. He also mentioned R. Hutner saying that one should "take it like a hot potato".
He did however say that one should not initiate a handshake even though it would be technically permissible.
He did however say that R. SR Hirsch did initiate handshakes with women.
I heard from a talmid of R. JB Solovetchik that he initiated handshakes with women. In one instance the women did not shake his hand and he said something like, "If you aren’t shaking my hand because you are angry with me, OK. But, if you aren’t because you think it is assur then do it."
As for the "dead fish" shake– the Rav, zt"l, gave everyone, including men, such a handshake.