I live in Arizona so my experience is probably different
but how do mature trees stop run off? in general ground cover - grass and other low growing, dirt-covering plants are what stops sheet erosion and run off. Trees don't actually protect the surface of the soil from run off much, though their canopies do protect soil from hard rainfall, hail, etc.
as to basement, can you pin point where water comes in? is it a leak situation, under a door/window or is is seeping through masonry or other construction? surface run off can be diverted if it has someplace else to go, but if it is seepage below ground level that seems to be a situation where underground barrier/waterproofing would be needed.
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