I have bad arthritis in my hands and I have found giving them my fingers and a very light feminine shake.
how to do it
when they reach to shake, grab their fingers have yours slightly bent, and place your thumb on the top of their fingers.
Do not place much pressure in the shake and most people will follow suit
I alway picture it much like in the way back when times when a lady would give her hand and the man kissed it. Well that is the hold I am doing for shaking.
Hope it helps and if its a really bad day then explain to them that your arthritis is acting up