There's no way to change that text size but you do have an option to have the npc dialogue in your chat box.
Go to Main Menu>System>Character Config>Log Window Settings>Log Filters.
Go over and pick the General log and a filter window would come up. Tab over to Announcement and you'll see two options, one is NPC Dialogue and the other is NPC Dialogue (Announcement). Make sure they are both marked, hit your Triangle button to apply the changes and you'll have the npc dialogue in your chat box.
EDIT: Forgot to mention. Some cutscenes will take away the chat box. Just hit Enter on your keyboard, which I really hope you have, and it'll bring back the chat box.
I don't have a keyboard just the PS4 controller as it's the PS4 version of the game. Would I need to move the chat box to the center of the screen so that it would display like the NPC dialogue box or will the NPC dialogue box still stay at the center of the screen? I can read the small text in the NPC dialogue box, but it sounds like I would need to make the chat box huge to get the text to display at the size I imagine in my head.