I'm coming back to classical after years of rock, blues, pop, etc. Muting in classical is something I never studied much, let alone mastered, so I've been attentive to it when I go to recitals and concerts. (Really, to all aspects of right hand technique, which deteriorated horribly during my time away.) So I've been freshly amazed at how agile and precise the thumb of an elite classical player can be around the bass strings, as it dances around sounding, muting, resting, hovering. That the classical thumb is expected to be as active and agile as any other finger is a recent realization, and especially with regard to precise muting, this is simply a facet of technique that's new to me and that I'm just going to have to buckle down and build from scratch.
A few observations to share with my fellow re-beginners, and beginners: 1) A string only has to be touched for an instant to be damped, so a quick thumb (or finger, or whatever), can do much in a fraction of a beat. 2) There are frequently opportunities for a LH finger to quickly touch a ringing string. 3) Some people can damp more than one string at once with their RH thumb; size, and the tip joint, can help here.