You could try putting something under the strings at fret one to stop any strings that arent being played from ringing out which is sometimes what i do. I usually use a piece of paper but paper doesnt work too well.
Playing slow and speeding it up slowly will help. It is a matter of making sure your fingers are not smashing the other strings from poor technique. Palm muting is another way but that also has to be practised. When you palm mute try and keep you palm muting hand still when your tapping so it doesnt hit strings as well.