We also have an old home (1915) and our closets basically used to be dead space in the house, so they are not well insulated or able to be better insulated, so they get pretty cold. We had an insulation company come to our house for many improvements but this was one that could not be handled in our home. I don't know how old your home is, you could call an insulation company and have them check to see if they could blow insulation into the walls of the closet. If you need the name of a company, send me a message and I can share the one we used. I can't remember it right now, but we were very happy with them.
Although our closets get cold, we don't have drafts in them. If you have drafts in your closet, you should definitely either try to figure out where they are coming from and fix them or hire someone to do this for you. If there is a window in the closet it may be as easy as caulking around the window and a handyman could definitely handle it if you aren't comfortable doing it yourself. If it is just that the closets are cold and the drafts you feel are just the cold air coming into the room from the closet, you can leave the closet door open. That way when you heat the room you will be heating the closet as well and it will stay closer to the temperature of the room.
Good Luck,
D.