How can I determine what type of washer/gasket to use for an antique shower?
I would just take one of the old ones with you to a home improvement or plumbing store and find a close match.

If you are trying to order them, you need to measure the outside and inside diameter along with the thickness.

Sometimes it is necessary to stack two of them if one set of threads is not that deep and doesn't make contact with the washer even when it is fully screwed in. In this case, adding a second washer ensures that the shower head makes contact with the washer and creates a proper seal.