Caulking around pop-up door?

Does the air from the door travel directly across to some exit point? I would think (depending on where you live) that a little air from below might help clear moisture and fumes if it exited at the top of the coop. Of course that assumes that your door doesn't face into the prevailing winter wind. Do you have a picture of your setup?

 

Yes, a picture really would help. How big is the gap? I couldn't tell where you're located...by your name I would guess North Carolina??? If so, you guys don't really get cold winters do you??? A tiny bit of cold air coming in won't hurt, especially since most pop doors are pretty far below the roosts. Now, if you're up north and get the really cold winters, and the gap is pretty big (but I doubt it's THAT big), then I'd worry more about sealing it.
How to seal it will depend on how big the gap is though...anything from pieces of quarter-round to window weather stripping may work to seal, depending on how the door is installed...

 

I'd be more concerned about the predator risk, myself. Is the gap big enough so a raccoon could get its paw in there and force the door open?

 

If it's a big gap - and I'm guessing it is since you're worried about drafts, then caulk isn't your proper material. That's really meant for small gaps. As mentioned, a photo would help. If it is a sizable difference, then could it be shimmed and covered with trim?

BTW, chickens will eat caulk if they can see it. I caulked first and then painted over it so haven't had as much trouble, maybe because they don't differentiate it from the walls? I don't really think that caulk will poison them, but still wouldn't want them ingesting too much of the stuff. Just thought I'd mention that.