Not sure exactly how you have it installed, but the simplest thing is *usually* to cover any exposed edges with a piece of wooden trim (square or quarter round in corners, any ole thing in other situations). Nail or screw, probably predrilling the holes in the trim unless it is really a bigger-than-normal-trim size piece of wood (thin narrow trim splits real easily if you don't predrill pilot holes)
I wouldn't trust caulk -- not only do poky things poke through it, but chickens tend to peck and eat it.
For larger mesh wire, you can individually 'kill' each stick-y-out-y wire end by bending it and hammering it into the wood there... but this is not a reasonable option for hardwarecloth, with so many so-relatively-thin wires. GOod for 2x2 or 2x4" fence wire though!
(e.t.a - if it is just a small area of exposed edge, another possibility is to take something like a screwdriver or messed-up old chisel, and hammer that against the edge of the hardwarecloth til it is bent down into the wood a bit. Covering with trim would be safer though. Doesn't even have to be wood trim -- anything metal or plastic you have lying around that will be rigid and durable enough would work fine too)
Good luck, have fun,
Pat