Buy a stud finder at Lowes. Don't get the cheapest model, it's too hard to use. Find two studs closest to the ends of the railing and insert screw in hooks to the studs. Hang your fencing on them. Item # 197656 | Model # SS E50
Zircon 1.5-in Scan Depth Metal and Wood Stud Finder
Use molly bolts. drill a hole into the wall, use molly bolt anchors specified for the weight of the fence.
Hi Holly: Your best bet is to use the studs behind the wallboard. If, perchance, your fence won't fit where the studs are, then get a piece of wood a little longer as your fence is, about 1/2 inch thick and about 3 inches wide, that will fit across some studs horizontally. Attach this piece of wood to all of the studs very securely, with screws that are at least 3" long. Pre-drill the holes for the screws. Then, you can attach the fencing to this piece of wood without worrying about it falling down. Oh yeah, paint or stain your wood before installing it:) Good luck