First we'll need to know what kind of begonia it is. Do you have a picture?

Begonia tuberhybrida.

OK, tuberous begonias-- let the little guys go dormant naturally. Then you can lift the tuber and store them (although some people leave them in the pots and bring the pots in). The tubers will sprout and start up again on their own time. At that point you plant them (if they are lifted) and begin to water some, give them light, and then if you have any issues with frost wait to set them out until frost is passed.

Thank you. We don't have any frost issues so I can keep this anywhere. Is it OK to cut off the top part of the plant, which is dying anyway and just keep the root or tuber part?

I would wait for it to die back naturally.

Where ever you store it, make sure it isn't in the rain-- you don't want it to rot while dormant.