What's the proper adjustment to prevent this toilet from running?
A test first before you do anything I have suggested. when the toilet is flushed and in the process of filling, gently lift up on the gray bar that holds the blue ball to see if the water stops. If it does, it needs adjusting. If it does not, it needs replacing.

There is a screw on top of the inlet valve that adjusts the point where the water stops flowing. Screwing it in while the tank is filling 1 turn should change the water level to what you need. 

Or try bending the gray metal rod that supports the blue ball so the  ball sets a  little lower than before, say about 1" lower, if you can manage it. Use both hands on the metal rod to bend it, the plastic valve that the rod connects to cannot take a lot of stress. That should stop the water flowing about 1", maybe less, to the top of the overflow tube.

If the water level still rises up and over the overflow tube, try the bend one more time, setting the blue ball even lower. If it still runs on, then the filler needs replaced, it is not stopping the water as it should.

Looking at the picture again, check to see if the blue ball is hitting the side of the tank, not allowing it to rise high enough to stop the waterflow. If that is the case, if there are visible threads, screw the ball in more to clear the side of the tank.