I started out in boots, and couldn't imagine wearing anything else. Then I tried approach shoes and now I only wear boots in the snow. I'm not convinced the whole "ankle support" thing is anything but myth. I can move much more nimbly in shoes, and they don't hurt my feet. Sometimes I just have a pair of thin casual socks (like you'd wear with a suit) and my feet don't complain once. Waterproof shoes will keep the wet out to a point, although obviously only up to the ankle. If you really don't like mud, boots might be the best choice.
I wear Scarpa Oxygen GTX approach shoes, and would unreservedly recommend them as go anywhere, do anything footwear. Long walks, scrambling, backpacking, even running.