Originally Posted by carldogs
For staccato notes especially over a strong harmonics like the 5th, 7th or 12th frets I use both hands to mute, if I have played a note with my index finger (right hand) I follow with the middle finger and stop it on the string to kill the note, the middle finger is already in place for the next note so I play the next note with the middle finger and stop the string with the index finger, I alternate like that for short notes as well as muting with the left hand as you are doing.
+1 It takes getting used to, but if you can rest your other finger for a split milisecond on the string before you play it (as opposed to playing through the string), that can help muting it.
Originally Posted by Sni77
+1 to this. What also works if you have some freedom in your left hand positioning: you can fret with your ring finger and use the index and middle behind the fret to mute the harmonic. Works great if you combine it with the right hand method described by carldogs.
If you're using upright bass-type positioning or chordal positioning on the left hand, you can use a barred index finger to mute stuff you might be playing with the middle, ring or pinky fingers. When you hit a note that you need to play with your index finger, you can mute the strings with the other fingers with a hammer-on sort of action, except you don't let the other finger hit the fretboard.
The point of all of this is that to eliminate that harmonics sound, you want to try to mute the strings in two locations at the same time. It takes coordination, and I've literally done repetitive exercises until it felt natural. Getting a really good, quick staccato sound takes some practice. Until it really feels natural, it's good to run your exercises or your scales playing very staccato and incorporating the above suggestions.
Once you get it good and down, it really starts to become second nature, and you don't even think about it at all while playing. Until then, practice staccato every now and then over different exercises.