Are there any 12V pins on a RAMPS 1.4 Board?

[printbus]

...I currently have the LED strip attached to the D9 fan connections, but the voltage there fluctuates as the fan speed changes and as the extruder & bed heaters get turned on and off.

A capacitor would only help sustain the voltage across a momentary and very short duration surge or dropout - like noise.

Do you have separate wires from the power supply for the 11A and 5A power inputs on RAMPS? If not, additional voltage drop in the 12V wiring when the bed heater turns on will affect the voltage available for the D9 output. If separate wires are ran from the power supply to the RAMPS 11A and 5A power inputs, voltage changing on the D9 output when the bed heater turns on and off has to be the power supply output itself varying with load.

The voltage on the D9 output varying when the extruder heater turns on and off could be voltage drop in the wiring to the RAMPS 5A power input, or could be the predictable voltage drop caused by the 5A polyfuse on RAMPS, which supplies power to the D9 output. The polyfuses have a finite resistance to them, and they work by temporarily opening up when their internal resistance has caused enough of a temperature rise.

A capacitor won't do much to help with either of these problems.

You do realize the D9 output, when used to control fan speed, is a Pulse Width Modulated (PWM) output? The voltage on D9 doesn't vary with fan speed. A meter might suggest it does, but only because the meter is displaying the average over a number of measurement samples. The D9 output is still only switched on and off. That switching just occurs very fast. For a fan, it basically gives the fan a burst of voltage to start it spinning and then shuts off to let it coast. Another burst of voltage and then lets it coast. etc. Fan speed is controlled by varying the width of the "power on" part of the pulses. Using PWM is a standard method of controlling brightness with LED strip lighting. If that's what you're doing with D9 and you see the LEDs flickering and you're using Marlin, it could be that the PWM pulse train is slow enough to be visibly noticed. A capacitor might help with this, but I'd start by uncommenting the FAST_PWM_FAN definition in Marlin's configuration.h file. That increases the frequency of the pulse train. If you're not using Marlin, the pulse train might already be fast enough or you might see if there's a different solution for adjusting the frequency of it. If you do add a capacitor across D9 to help average out the pulse train, unless it ends up being really large there's likely enough resistance in the wiring, the polyfuse, and the capacitor itself to not damage the D9 MOSFET. Size of the capacitor would depend on the current needed by your LEDs. Trial and error would likely be the best way to size it.