“Is it allowable for me and women [who are eligible to marry] to shake hands sometimes?”
“The origin [of this ruling] is that a man is not allowed to shake hands with a woman [who he is eligible to marry and vice versa]. However, if she is an elderly woman then the school of Abi Hanifa [may Allah have mercy upon him] considered shaking hands with her permissible. Perhaps, the questioner will state that, “By not shaking a woman’s hand it could lead to hardship or shared ill feelings and so forth?” If not shaking hands will lead to the actuality of those fears, then it is allowable to shake hands as long as there is no evil feelings or affection for the woman [and vice versa]. And if the opposite holds true, then it is not allowed to do so.
Dr. Abdullah bin Bayyah
Note, there was some abridgment done to this answer to make it clearer to the English audience. However, the meaning and answer of Dr. Bin Bayyah was not changed.