From The Ask the Teacher Section at Rock House (http://www.rockhousemethod.com/) comes some good advice from John McCarthy.
Q: When playing scales, as I play through them I can still hear the previous notes ringing out. I'm wondering how I can mute that noise after the notes are played. I have the same problem when playing arpeggios.
A: First of all you need to know how you should be playing scales and arpeggios.
Scales and Arpeggios should be played one note at a time and without any two (or more) notes ringing out together. Try to make the notes blend smoothly in a row holding each picked note down until you pick the next. There should not be a sound break between notes this would make a staccato sound (short, detached, choppy).
If the problem you have is open strings ringing out, this has to be cured with your right hand or picking hand. Try to control the strings that you are not picking with the palm and side of your picking hand. This muting technique is done where the stings come off your bridge; try not to move your hand too far away from the bridge because it will deaden your sound completely.
Experiment with these techniques and you will find the perfect way to control your sound, remember practice makes perfect!!!
Hope this helps
Yours in Music,