There is generally no harm in running a fan or any DC motor on lower voltage.
If the voltage is too low, then the motor will not run (or continue running if already spinning but not start), because there is not enough torque generated to oppose the “cogging” effect of the layout of magnets inside the motor. (This effect is why when a fan is turned off you will see it bounce and stop at a particular angle, and you can feel it by pushing the blades around). In this case, the motor is stalled and in principle this could burn out the motor just like any other stall, but that is less likely as the voltage and therefore the stall current is lower.
If you find that a fan promptly starts up at a given lower voltage, then you should feel free to use that lower voltage.
If you need a fan to operate at a wide range of speeds including very slow ones, then rather than changing the supply voltage, it is better to use a fan which has a PWM (pulse-width modulation) input, which is driven by a microcontroller or other oscillator (e.g. a 555 chip) you provide and tells the brushless motor controller inside the fan what speed to run at. Because the controller has access to the full supply voltage when needed, it can start up and run at much lower speeds.
Since you're interested in climate control, this might be a good future project — a temperature-controlled fan. It can be done with a single-purpose fan control IC, a microcontroller, or even with separate chips (an oscillator and a comparator at minimum).
Finally, I'm guessing that you don't have this problem, but just to avoid any confusion for other readers: from context, when you say "LED power supply" I assume you mean a regular (constant-voltage) 9 V power supply, not a constant-current LED driver. You should not run a fan off the output of a LED driver.