Position yourself at one end of the box spring and have a strong friend stand at the other end. Make sure you're at the narrow ends of the box spring, that is: At the head and the foot, not the sides. The person who's willing to walk backwards and go up stairs backwards should be nearest the stairs.
Bend from the waist, squatting to lift with your legs, as you grasp one edge of the box spring. Have your friend grasp the same edge on the opposite side, then, acting together, lift the box spring up. It should now be vertical, or perpendicular to the floor.
Use one hand to stabilize the upper portion of the box spring, either at one side or from the top, depending on the length of your arms and the size of the box spring.
Walk slowly toward the stairs, one person moving forward while the other walks backward. Take your time and don't rush so that you can synchronize your steps as much as possible. It's the responsibility of the person moving forward to let the one moving backward know when he is getting close to the stairs.
Move slowly up the stairs, keeping together in tandem step as much as possible. You can also have the person on top move up one step first, then the person on the bottom move up a step, pushing the box spring up toward the other.
Lower the box spring to the stairs together if you need to rest. It's the responsibility of the person on the bottom to keep the box spring from sliding back down during these rest stops; usually this involves nothing more than just leaning your body weight against it.