Of course, latitude also plays its part. If the valley is around, say, the arctic circle, then perhaps being a couple miles deep would bring it up to a nice sub-tropical warmth. Its a nice idea, really, but my first question (as a world builder that strives for realism) is
why is this massive valley present? I can think of two good (and non-magical) reasons off hand:
1) Collapsed magma chamber from a supervolcano that erupted millions of years ago. There are regions on the earth today that - while not so deep as that - are thousands of feet deeper than the surrounding region due to this. Eruptions at such sites tend to occur every half million to million years.
The depressions formed by such are not even necessarily round regions. Look up "Long Valley Caldera" on google (the first link to appear is
this one. Its an oval valley about 15 by 30 km and about 150 m to 1000 m deep (depending on which end you are at). However, the entire thing is about 6500 m above sea level, thus the reason it is not a contestant for deepest surface on earth.
Just take this, and multiply it about five to ten fold (in size) and it will be what you want.
2) Rift Valley. Occationally the crust bulges and then rifts, usually in a Y shaped pattern. Consider the Rift Valley in Africa and the Red Sea (two prongs in that Y-shaped Rift). The region - when it cuts into a continent - is often steep-sided and relatively flat. Parts of it will have hot springs and bubbling mud flats due to geothermic activity, and earthquakes are about as common there as one might expect to find in California or Ice land or Japan, to use some examples. There are also likely to be some volcanoes nearby - but not necessarily in the rift itself.
Suppose such a rifting occurred not next to a continent, but in the center of one? Suddenly we have a vast Y-shaped crevesse in the center of a continent, slowly dividing it into three continents. Now suppose we are near the arctic circle, and the rift occurs in a vast plateau. Now we have a rift that - if deep enough - provides (warm?) temerpate (rather than sub-arctic) temperature throughout the year, and is surrounded by very high steep slopes that likely have volcanoes dotting its edges - or forming on the outskirts of the plateau beyond the immediate border of the rift.
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Now in regards to your deep valley, note that depth does not directly affect humidity or percipitation. If the rift is in a dry region, then likely there will be little humidity, while if it is in a wet region the humidity will be as terrible as might be found in a tropical swamp.
If you go with my idea of having it near the arctic circle (not above it, as that would cause several months of darkness, which would prevent trees and other temperate / tropic plantlife) then it is likely that during the periods of snow any snow that falls into the valley will become rain by the time it hits the valley floor. Furthermore, if the valley is deep enough, it may have its own clouds forming, perhaps where the temperature meets some critical balance between the chill of above with the warmer below - forming vast mid-air fog/cloud- banks that would percipitate to the valley below.
This is actually a very nice idea. I might have to make use of it myself.