I'm slowly making progress muting strings. I find that both hands come in to play depending on what you're doing. Sometimes it's picking hand palm muting, sometimes it's picking hand fingers touching the unused strings, sometimes it's fretting fingers that are being used to fret a note on an adjacent string, sometimes it's unused fretting fingers on different strings.
I find string muting to be the biggest unexpected challenge in playing a decent guitar that has some resonance and sustain. Dead guitars that play like clubs don't usually present muting problems.
Which strings to mute? That's easy. The ones that are creating that dull roar that should not be heard behind your picked notes!
I wish you the best in your string muting efforts. It's not easy, at least at first. But it is possible.
Have you tried Lesson 10? For me, that's a string muting nightmare. I still don't have it quite right but I'm getting better.
See you down the road!