How to prevent cockroaches climbing onto something?
I had a client who had the same problem with the box bed. I put two wraps of foil tape around the frame and it kept those little bast...ds from climbing up. You have to make sure the surface is really smooth and kink free. You can pick up this tape at any home store. 

Also, try to bug proof your place. Check for openings around drain and water pipes. Place a few of those "Roach Motels" around the place to kill off some that might already be in. Other than spraying insecticide, which many people don't want to do, I don't know anything else to do. There's a reason they've been around since before the dinosaurs and have survived. Good luck.