It's your house, you can do whatever you want. Just kidding. I just finished a basement remodel in a upscale home. The homeowner wanted painted woodwork because the rest of the house had it. The size of the woodwork was scaled down slightly. The upstairs had 9 and 10' ceiling heights. The basement was 8'.
In my own home, I have stained woodwork on the first floor (ranch). I stained the wood trim in the basement when I finished it but used all oak downstairs with a golden oak stain. First floor is poplar stained to look like dark cherry. Whatever suits your fancy.