http://www.byond.com/developer/Leur/ ScreenResolutionRetrieval
It uses Javascript and a hidden browser with client/Topic to retrieve the clients screen resolution.
Or if you don't feel like downloading the demo, and only want to know the basics of it:
//snipped//
src<<output({"<script type='text/Javascript'>
document.location.href = 'byond://?action=resol&width=' + screen.width+'&height='+screen.height;
</script>"},"resodet")
//snipped//
client
Topic(t)
var/nlist[] = params2list(t)
if(nlist["action"] && (nlist["action"] == "resol"))
src.mob.scrwidth=text2num(nlist["width"])
src.mob.scrheight=text2num(nlist["height"])