Think of a motor like a resistor. As you increase voltage, current goes up. Of course, a motor is not a simple resistor because it's resistance changes with frequency, load, etc. But the analogy works for this.
Now, if you gear down a 6s setup to have a similar speed as 3s, then current will be reduced because gearing down reduces the load on the motor (effectively increasing the resistance).
Also, you might find yourself applying less throttle on 6s than on 3s, so that may help too.