Yes mice can and do climb up counter tops so if you have any food that the mice could get into you should put it up away from the mice especially if you have mice in the house.
You should also set out either glue traps or snap traps for the mice on the counters too so that when they climb the counters they'll be enticed to the mice traps.
I catch the majority if my mice on my counters even though I sometimes put traps on the floor near the counter it seems they run across my counters more looking for food even though I keep my food off the counter when not eating it.