I finally found this story after more than 25 years. :) The story is called 'The Cocoon' by John B.L. Goodwin. It appeared in one of the Alfred Hitchcock Presents anthologies (Stories for Late at Night). I just finished reading it downstairs in the chilly living room, with the curtains open. Now I'm back up here where it's warm and the curtains are closed. I hope nothing starts flapping against the windows. ;)