Filtering the heck out of the supply line is a first easy start.
Try a 25V, 1000uF across the terminals at output. And placed very close to the output of the supply place a 0.1 ceramic capacitor and makes sure the leads are very short. Then at the output (+12) _after_ the newly added ceramic, place in series 100uH inductor and make sure it is rated for a current of >1 Amp (like a wire-wound toroid with low DC res.). Then place the 1000uF. Then take this output with which will now be the +12V filtered and run that to your device.
If this doesnt solve your problem, you may be getting pickup due to the long cable length and big supply loop created. In that case, you should try shielded cable or twisted pair.
+12(noisy) -> Ceramic cap -> inductor -> 1000uF electrolytic -> +12(cleaner)