Pretty sure Brian uses the side of his palm that his little finger is on to mute the bass strings. That would be the standard technique, otherwise it would interfere with picking and strumming. I might also use the little finger of my picking hand to mute the high E string on occasion. The left hand can help greatly with muting, also. The thumb over the fretboard can mute the low E and, potentially, the A string. The tip of the barring finger can just touch and mute a bass string above. You can lay your index finger down across the strings to mute all the strings below the string you are picking, SRV style, The third finger can mute the high E string on an A shaped bar chord and, of course, you can release pressure on a bar chord to get muted strums. Muting strings really is a two hand technique.
John

I find muting so difficult that I rarely use it. Either I end up muting the string i’m playing, or some of the strings just ring out of control. Muting is essential for raking, which is a good dynamic effect, but I just can’t get control of muting so I don’t do it and just sacrifice raking effect. Maybe i’m just too lazy to really work at it, but it doesn’t come natural to me at all.

I kind of use a combination of my two hands, whichever feels right to me. It’s generally easier for me to use fingers of my fretting hand to mute strings. I do use the outside pad of my picking hand as well. I don’t like to overthink it. I’m not great at it but seem to be improving. Sometimes when playing around and muting strings, it just comes natural. Other times, not so much.