Wait until they go out of flower and begin to fail. I usually let the pots or containers go dry. when dry remove the corm. Make sure there are no vine weevil grubs on board. Store in a frost free place. This year I had only one corm left viable after the severe winter. I lost a huge one I have stored in a hanging net for years. Next year I will buy new ones and take extra measures if the winter goes really cold.

I would store them in a box of sand or completely dry compost, cool place but free from frost or mice.

The answers above assume we're talking about tuberous begonias.If they're the smaller, fibrous rooted sort just buy new ones. I've tried keeping them over winter without success.

I've kept a single fibrous rooted one in the house as a pot plant but that's hardly practical for six basketfuls!