Have you checked the temperature variation in an interior closet? If you keep the door closed, you shouldn't see swings like you do in a big room. Get a digital thermometer that records and saves hi/lo readings and see if you have much change. If it's going to stay in the basement, do you have any warmer spots, say near the heater? You could also build a little box out of rigid insulation or styrofoam that you could put around your carboy.
A carboy full of fermenting beer has a lot of thermal mass, so it doesn't swing around in temperature as much as the surrounding air does. If you add another level or two of buffering to those air changes, the air next to your carboy won't change as much either. Also keep in mind that a carboy of fermenting beer will throw off heat as it ferments (fermentation generates energy), so if you do insulate your beer, it will keep heat in as much as keeping temperature change out.
Worry less about the air temperature and more about the temperature in the fermenting beer. That's what affects yeast performance.