As far as I know, it is tied to the size of the font. (Unless something has happened in your game that I don't know about.) It's also the height of selected text if you select text. You can see it similarly when editing here. The reason it might seem too tall is that fonts have ascenders and descenders that the overall font size leaves room for.
I was going to show some examples of characters that have bits that high, but I can't seem to actually find characters that exhibit that. I tried some with umlauts and things, but I haven't found one yet that gets anywhere near the top of the selection box. Part of it might be that the spacing between lines is factored into the overall font height, but then I tried changing the "line-height" CSS attribute, it even began clipping into the text, but it never changed the caret height.
But everything I've seen googling around looks like the answer is "no".
Thanks for looking - I've had similar findings.
I managed to reduce it slightly by changing the height property as you did, but the smallest I could get without clipping into the font was 18px so the difference was hardly noticeable.
I'm adamant Quest's caret is larger than most. Look at the caret used in Inform/parchment games (http://playfic.com/games/cooper/tutorial) relative to text size, and compare it to Quest's (http://play.textadventures.co.uk/Play.aspx?id=a7bbyvr5ik262bqo1vgfag)
If it is my doing, I've no idea what I've done.