my sister made a complete cover for a similar bassnette. she used pre quilted white fabric. It was all one piece, inside was made just like you were making a big basket then she added about 2 ft of same fabric completely around the outside for the skirt. She made several big box pleats. She added a simple row of light blue ribbn around the bottom with decorative stitching in blue.
you could leave the ends open to fit over those handles
her grandson only stayed in it for a month before he moved to a regular crib.