It is best to let them totally die off. Less time you have to worry about storing them also. But watch for moles, as they can really hammer up a potato patch. A lot of rain , can also rot potatoes if soil is heavy and poorly drained.
Unless they are New potatoes (a variety of potato often grown in the MidAtlantic states) let the stems and leaves die off. New potatoes are the first to be harvested but have a very short shelf life. They will rot in the ground if you leave them until Oct.
When we grow potatoes, generally russets, we dig as needed from about labor day until the beginning of October and then dig the rest.