After you cut the fruits off stand them, stalk down, on flower pots to finish ripening. Mine were all beautifully orange this year.
If the vine itself is still alive and the weather is mild enough, I'd leave the fruit on for as long as possible but use some support underneath each fruit to lift it off the soil.
If the vine is finished, then Kathryn's advice is good. You just have to remember that there is a difference between the fruit ripening and it "curing" ( which will allow you to store the fruit for longer periods )