I prettied up the inside of the changing table with fabric drawer liners. Here's the how-to if you'd like to make your own.
Double stick tape
Iron and ironing board
First you will need to stiffen the fabric. You can by premade fabric stiffener, but Elmer's glue works just as well. In a large bowl, mix equal parts glue and water. Soak your fabric in the glue mixture until it is completely coated. Ring out any excess glue mixture. Lay flat to dry. When it is nearly dry, iron out any wrinkles. Your fabric should be stiff like cardstock.
Create a template of the bottom of your drawer using newspaper. Trace your template onto the back of your fabric and cut out.
Place a piece of double stick tape in each corner of the drawer and lay the fabric on top, smoothing it out and pressing down firmly in the corners so it sticks to the tape. You can also use spray adhesive to secure it to the drawer, which is more permanent than tape. Just spray the adhesive onto the back of the fabric and smooth it out onto the bottom of the drawer.