|Are you SURE you're pressing the right button ?|
If it has a plastic button that presses on a switch behind it, it may be stuck inwards in the monitor case - if you un-stick it, pressing it should work fine.
If pressing the button doesn't work, remove the AC power to it - unplug it or switch off the power bar it's connected to.
Or - if you want to have the monitor on and the computer running all the time, which is NOT a good idea if you don't NEED to do that, and your problem is it's always displaying, RIGHT click on a blank part of your main desktop screen, choose Properties, then click on the Screen Saver tab, then the Power button, and set the monitor to turn off (go black) after a specific amount of time.
Your display will come back on automatically when you move your mouse or press a keyboard key.