Generally this happens when you're using a noise gate or downward compressor to compensate for a bad mic. They're just there to make the problem less obvious when everything's quiet, with the assumption that the sound of a voice will drown out the self-noise hum.
You can try using a noise-reduction filter (though those tend to have... less than pleasant artifacts of their own).
Really though, it sounds like either you have not-great microphones/amps/mixers/interface (fix: buy better equipment once you track down the offending piece of gear), or a gain-staging problem somewhere in your audio chain (fix: gain stage your setup properly; you want to set your mic amplification ONCE, as close to the mic as possible to keep the noise floor low, and run everything downstream at unity).
I'd suggest trying to gain-stage the setup first (if nothing else, it's cheaper). There are tutorials on YouTube on how to do it. Also, apply duct tape or superglue if necessary to keep people from playing with the knobs. Gain is not a volume control, and lots of non-audio people treat it as one... especially on belt-packs if they feel they aren't loud enough. Which results in a high noise floor, and hum.
