Nail the shelving into the box, spacing the bottom three shelves just far enough apart to fit baskets with a 1x8 board below them.
Attach the shelf base to the bottom of the basket using construction adhesive and two or three 1/2-inch screws. The back of the basket should overhang the back of the shelf base by about 1/2 inch. The front of the basket will overextend the shelf base and the shelving unit itself by a few inches.
Screw the other half of the drawer runners onto the inside of the shelving unit. These should sit directly on the bottom three shelves, and the front half should sit flush with the front of the shelving unit with the back half butting up against it.