Rav Moshe has three Teshuvos on shaking hands with women. He forbids
initiating a handshake (which, indeed, all, or virtually all, Poskim forbid), but is Melameid Zechus on those who are lenient to
return a handshake, concluding, however, that such leniency is difficult to rely upon. That falls short of saying that the practice is Ossur, although Rav Moshe does not say it is Mutar either. (Clearly, Rav Moshe would not hold that the practice is Yeihareig v'al yaavor. Indeed, when I spoke to Rav Reuven Feinstein about this topic 2 years ago, he dismissed the possibility that it would be Yeihareig v'al yaavor, stating "how is it shayich Yeihareig v'al yaavor on a Machshava?").
The 3 Teshuvos are summarized here:
www.aishdas.org/avodah/vol10/v10n045.shtml#01 (and, yes, I did learn them in the original).
Part of the problem with being lenient is that some people won't know where to draw the line. Some people have foolishly suggested that, nowadays, a kiss on the cheek or a hug are not Derech Chibah v'Taivah any more than a handshake is. So, one who is lenient has to be careful not to allow shaking hands to become a slippery slope.
I was once at a goodbye party at work, and the female honoree wanted to hug me. I felt my entire eternity flashing before me, and I recognized that I was dealing with a genuine case of Yeihareig v'al yaavor according to all opinions. I simply responded "I can't." "Why," she asked, "do you have a cold?" "Yes," I responded quickly (I didn't), and that was the end of the matter.