When I build a fire in the upstairs fireplace, the basement fireplace smokes

This is a common problem that is due to air pressure issues in your home. If you have a well-insulated house, the most accessible route for air to enter the structure is often the unused fireplace chimney. As the air is drawn down the flue, it picks up smoke that is that is exiting nearby from the fireplace in use and expels the smoke into the living area. A solution to this issue is to provide makeup air to the house, so the adverse pressure problem no longer exists. Another solution would be to have a top-mount damper installed on the fireplace that is used the least in your home.

Source: Chimney Sweep Raleigh NC | Chimney Sweep Charlotte NC