But seriously, I approached this with a thought experiment.  Replace the water with something solid.  So imagine instead the glass is filled with cement, when you push your finger down you obviously end up exerting a downward force on the cup through the concrete and thus the scale would be moved.

Now I'm trying to think of any property of water that would prevent the same thing from happening.  Think of it this way, when you push down on the concrete, the concrete is trying to move away from your finger but is prevented from doing so by the cup, thus a certain amount of force is exerted on the bottom of the cup and it is this force that results in a downward motion of the scale.

Now I imagine the same principle holds with water, the water being displaced by your finger causes a force to be exerted against the cup (both the sides and the bottom), the force exerted on the bottom would cause a downward force and thus in a perfect world would result in a downward motion.  Of course real world things to consider is that it takes a certain minimum amount of force before the scale will move and it is to be determined if the force from the displacement of water exceeds that minimum.  I imagine most likely not.  

So all that being said, my vote is on the scale moving downward, only other thing I would consider is the fact that you are holding your finger in place and thus your arm is exerting a certain amount of force in an upward direction to counteract gravity and perhaps this force would negate the force from the displacement of water.