One of the things that is soon obvious when playing the guitar is that sound is easier to start than to stop. We play an open string, but then our attention is diverted and we attend to the next notes we need to play, forgetting all about that open string, still ringing in the background. The harmony changes, but that note is still there. This frequent cushion of unwanted sound undergirding the music we play interferes terribly with our desired result, and yet is relatively easy to fix. It's surprising how often even advanced players are careless about dampening open strings. This article will describe the various mechanisms at our disposal for dampening.
I have two basic, go-to techniques for stopping bass notes. The first guarantees a legato articulation as it permits a tiny overlap of sound. Play the first bass note, play the second bass then immediately put your thumb back on the first bass note. There, it's been dampened. Of course this is not so convenient if the bass is fast-moving, or if the thumb has other notes to play also and doesn't really have time to go "back."
The second most-used technique only works on lower basses. It is used for both simultaneous dampening and cutting the first note off before the new one, introducing a silence between them.
After playing, for instance, the 6th string, and as you prepare to play the 5th, lower your wrist some, enabling your thumb to lay flat across both basses as you prepare and play the second note. You can adjust the contact minutely to change the timing of this dampening move, enabling lots of different articulations. This technique is not restricted to just lower adjacent basses; it can be used two strings or even three strings back, but the farther back the string being dampened, the flatter and lower the wrist needs to be. I refined this technique by practicing Sor's Study Op. 31 No. 20 (Segovia's No. 9). Learn to simultaneously dampen all basses while preparing each chord, creating perfect silences on the rests. Doing so requires the low wrist I'm referring to.
Of course guitar music, in all it's variety and complexity, requires more dampening techniques than these two. But these two will cover most circumstances. We fill in the gaps with a range of additional tricks, including dampening with right hand fingers other than the thumb, dampening with left hand fingers (very common and useful), and laying the entire right side of the right hand down on the strings for a full 6-string mute.
So far, I've been referring only to open strings. When notes that are being fretted need to be dampened, it is easiest and most effective to simply release pressure on them. This must be done gently though, or you may excite the open string, creating new problems. I have heard this technique dismissed, to my surprise, by at least one professional guitarist, who asserted that all dampening should be done with the right hand. I couldn't disagree more. I find it's easier to get a perfect conclusion to a note with this simple release of pressure than with any other method. (I have seen players who had trouble with this though: if the fingertips are sticky/sweaty, they can inadvertently rearticulate the string when lifting off, thanks to being, effectively, stuck).
To me, over-ringing, unwanted notes are like a musical heigeine problem. When I hear a student playing with lots of this type of noise, I imagine their unkempt apartments, chunks of spinach in their teeth, their shirts mis-buttoned. The musical effect is dirty.
Unfortunately, like many other elements of playing, it is FAR more difficult to train the hands to solve this problem after a piece has already been learned than it is to incorporate proper dampening up front. So, I recommend that this issue be addressed as a piece is first being learned, and the piece will never have unsightly spinach in its teeth!