You know, it depends on the situation. I have worked hard on controlling muting. If you use a lot of gain, this becomes important, but also if you do a lot of right hand up and down, a la SRV.
If you lift off the left hand fingers, you need to mute strings more with the right hand. If you leave them on (but not fretted), you usually have more control over the muting/string ringing issues.
Not sure if that reply helped, let me know if I was unclear.
I have good control when it comes to muting, and it just happens without me thinking about it, because I was a shredder in my youth, and thus learned to control string noise and that has helped me to sound more "clean" when I play today.