I've done something like this with audacity, but not a good while to remember exactly how.
Some 6-7 years ago I was excited about starting a business, and was playing with building a call IVR system that I recorded my own voice audio for voice prompts, but picked up the rest of crap in my house too. I used some basic noise filters through standard installed filters to remove stuff almost brainlessly. I remember a buddy of mine checking it out for me, he asked me if I did these professionally edited or something, and was just like "nah, me screwing around", but it ended up being really clean audio recording through my Jabra 510 speakerphone.
For your use case with video, you'll probably need to extract the audio stream, run it through audacity with the noise filter, re-mux it back in. I don't deal with video editing really to tell you the best method there, but probably containerize it as a mkv. I'm sure you ought to be able to mux the audio/video feeds with existing tools, mostly what folks do with video all day for pira^H^H^H^Huploaded videos all day.
HTH!
-mb