Calculate gingham dimensions: Measure circumference of basket opening, and add 1 inch (for 1/2-inch seam allowance). Measure from the top to the bottom center of the basket, and add 3 1/2 inches for fold and hem allowance. Cut the resulting rectangle on the bias.
Fold gingham in half horizontally, with right sides of the fabric facing. Sew short sides together. Press seam open.
Cinch the bottom: Double-thread a needle and hand-stitch along the bottom edge with a loose running stitch. Pull thread to close the bottom, and knot thread.
Fold top edge over 1 inch, and press. Position fabric in basket so the cinched section is centered in the middle of the bottom. Drape top over the rim of the bag (it should gather and pucker around the handles).
To hold the liner in place, hand-stitch along the pressed edge, through the fabric and the basket fibers.