Drums, as most wooden instruments will get effected from changes in tempature and humidity.
This does not mean that your drumset will fall apart in a cold basement. Extreme differences in temperature (like real hot days and cold nights) will cause the shell or the finish to crack and swell. Extreme humidity will cause the chrome on the hardware to pit and your shell to get damaged.
Just how cold is your basement? If it's freezing cold, I wouldn't keep my drums there. But if it's reasonably cold and not DAMP (very important) you could keep your drums there. Just check them once in a while and if you see any cracks or anything take them to a warmer place (not near any heat sources!)
