Can I add a second thermostat in my basement for more control?
The other issue will be that if the temperature sensor is in the basement, the upper part of the house may get too warm by the time the furnace displaces all of the cold ait in the basement. Remember, warm air RISES, so if your return air resister is not in the basement, it will be a long time before there is enough air exchange to make a difference down there. 

What I would do (have done) is to add a baseboard heater as a supplemental system, with its own thermostat in the basement, along with a way to keep the basement air from rising out of the basement (i.e. stairwell door).