Measure the width of the wall hanging or tapestry. Add 2 inches to this measurement, or more if you prefer the hanging rod to project from each side of the hanging.
Cut a dowel or order one cut to size at a hardware store. Cut the stiff hook side of a length of hook and loop fastener to the same length. Removing the backing from the fastener and apply the adhesive side to the rod. Press it against the rod in a straight line, using your hand to squeeze the fastener securely to the rod so that it adheres to the curved surface.
Extend the soft loop side of the fastener along the top edge of the tapestry. Cut off the excess. Sew the fastener to the tapestry by hand or by machine.
Press the fastener to the rod, matching up the hook and loop closures. Have an assistant hold the tapestry up by the center of the rod in the desired position. Stand back and check the position and have your assistant raise or lower the rod if needed. Aim to hang the tapestry so that the center of it is 6 to 12 inches above eye level when you view it from the landing. With an abstract design or large pattern you have more latitude and can hang it from the ceiling if you prefer.
Set a carpenter's level on the rod to get it level so the tapestry will hang straight. Mark the desired hanging location on the staircase's blank wall in pencil at each end of the tapestry. Screw in the brackets or hooks directly to the outside of the pencil marks. Rest the dowel on the brackets or hooks.