The drawers of older desks and dressers often ride on wooden runners, not slick metal slides. This simple, wood-to-wood friction system works well enough, but over time the bottom edge of the drawer sides wear out and the drawer starts to tip down.
Rather than replacing the runners in the cabinet and splicing new wood onto the drawer sides, try this shortcut: Remove the drawer and press a large thumbtack down into the face frame on each side of the drawer opening. Slide the drawer back in so that it now rests on the thumbtacks. If it's still sitting too low, add a couple of thin cardboard shims under each thumbtack to raise the drawer.