To keep the parts of the bookmark from flapping open, you will need to use fabric glue, fusible interfacing, or a similar product. Follow the manufacturers instructions for the product you choose.
If you use glue, use a small amount. Too much glue will soak to the outside of the fabric and mar your design. It may also cause the fabric to curl as it dries.
If you do not want the edge of the fabric to show on the back of the bookmark, cover it with a small piece of ribbon.