Cut tops from beets, leaving about 1 inch of stems.
Cook beets in 3 cups boiling salted water about 20 minutes, or until beets are just tender.
Drain; reserve 2 cups of the liquid.
Cool beets slightly and slip off skins.
Cut into thin, crosswise slices.
Combine salt, sugar, and cornstarch in a saucepan.
Gradually stir in beet liquid and vinegar.
Cook over low heat, stirring constantly, until smooth and thickened.
Add beets and apples.
Cover and cook over low heat about 15 minutes, until apples are just soft.