I've had that idea before too... it seems very tempting to get "free" cooling by simply moving air from the cooler basement to the hot upper floors.
I've never tried it, but some problems with this idea might be:
- It takes energy to move air, which will turn into extra heat in the cold air. Since you have to raise the cold air several meters, you'll be pushing against natural convection. The temperature coming out of the top of the stack will be somewhat warmer than the temperature in your basement.
- Since the temperature difference between the hot and cold sides isn't that great (think about how cold the output of an air conditioner can be), you'll have to move a lot of air to make any sort of noticeable difference. This will make problem #1 worse - moving more air means more energy.
- Any air you remove from the basement will have to be replaced by air coming in from somewhere. This air will be warmer, and depending on your climate, may have significant humidity. When the warm air comes into the basement and cools off, its percent relative humidity increases. This process happens naturally all the time, but you'll be accelerating it with your fan. If you already have basement moisture problems, this will make them worse - and it might make moisture problems happen if you don't already have them.
Oh, and one other important thing just in case it applies to your particular situation: in the very unlikely event of a fire in your basement, this air duct could easily become a chimney that pipes hot combustion gases straight up to the upper floors. As those hot gases leave, fresh cold air will tend to rush in and feed the fire, making the fire worse. This is why building codes typically require firestop protection in basement ceilings around fire-prone areas (such as your furnace). Definitely don't do this if the inlet for your duct is near your furnace or any other possible source of fire!