On PS3, you're able to move the dialog box where you want and it will stay there (except during cutscenes, when it always reverts to its default location). On PS4 this is pretty screwed up, but with a bit of trial and error I was able to get it to somewhere close to what I wanted. It doesn't seem to correctly remember the location you gave it, so on the next dialog it will place it further down or towards the edge. If in your case, it's going too far down, then try dragging it to a position higher up than you tried setting it last time, even if that's also higher than where you want it. It may take a few tries to find the magic spot to place it which it will interpret as going where you actually want it the next time. (And if you want pixel perfect control over where it goes, you're probably out of luck.)

It's easiest if you find a random NPC with a single window of dialog that's not related to any quest, so you can bring the dialog up and dismiss it over and over while you experiment with positioning it.

(In my case, I wanted to move it to the bottom left corner, and it kept coming back up as half off the screen. I eventually found a spot a few inches to the right and about an inch up from the bottom, that made it come back up in that bottom left location I was trying to place it in the first place.)