Praise be to Allah.
It is not permissible to shake hands with non-mahram women at all, whether they are young or old, and whether the man who is shaking hands is young or old, because of the danger of fitnah which that poses to both parties. It was narrated in a saheeh hadeeth that the Messenger of Allaah (peace and blessings of Allaah be upon him) said: “I do not shake hands with women.” And ‘Aa’ishah (may Allaah be pleased with her) said: “The hand of the Messenger of Allaah never touched the hand of any [non-mahram] woman, and he used to accept their oath of allegiance in words only.”
It makes no difference whether shaking hands is done with a barrier in between or no barrier, because of the general meaning of the evidence, and so as to block the means that may lead to fitnah.
And Allaah is the Source of strength.