Some stereos will take a feed from the light switch to change the display brightness at night. However, to be honest, it sounds like you have an actual issue with your wiring - all vehicle electrics should be at ~12-14 volts, anything less indicates a problem with voltage drop. Often a bad ground, a bad contact or even a short.
Remember, also, that current isn't restricted normally - the device will pull whatever power it needs, and if the wiring is designed for low current, then there should be a lower rated fuse to blow. As you're not blowing fuses, it sounds like a voltage problem.
EDITED: 12 Nov 2013 10:05 by PILOTDAN