Can you get herpes from holding hands with someone who has a cold sore if you have a damaged cuticle?

Unlikely possibility. Herpes is a virus theoretically possible to transmit but practical aspect no. Casual hand contact is not the usual way , also hand washing important in control of several types of infections , so always wash hands if you touch anything.
Not likely. Cold sores are caused by the herpes simplex virus (HSV). There are two types of herpes simplex virus: HSV-1 and HSV-2. Both virus types can cause sores around the mouth (herpes labialis) and on the genitals (genital herpes). The target cells for these viruses are confined to those areas, not your hands. The usual transmission is through kissing and sharing utensils.