I'm an intermediate player, and I just started sweep picking. Even on three string sweeps, my strings start ringing, especially when I bring my pinky finger to pull off a high note on the 1st string. Usually it is the G string that rings the most with a reasonable amount of gain. Let me know if you're having this problem. 

lukethrash7506 A better sub-forum for this discussion would be https://www.ultimate-guitar.com/forum/forumdisplay.php?f=9

Yes, it is an issue which is solved by practicing. You have to mute the strings your are not playing with the palm of your picking hand. Live, you can use something like Fretwraps to help you out, but do not let them become a crutch when you are practicing.
You have to practice slowly and fit in the action of unfretting and muting the previous note simultaneously while fretting the new note. If you for example mute your strings and hit them you will just get a muted sound, now while sweep picking you need to effectively do this on every string other than the one you're playing, or more realistically the string above and below the string you're currently playing needs to be muted in some way so that only the picked note is ringing out.

It just comes through practice, start slow and then speed it up gradually to maintain clarity. Don't force it, it won't happen over night and at best you can expect the process to take a few weeks. Also check out Troy Grady on youtube, if you already haven't.
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lukethrash7506 As Bigbazz says, you need to use muting.  It is possible to make a "tunnel" with your picking hand, so that bass strings are muted by the little finger side of your palm, or by your thumb, while the treble strings are muted by the top joint  (nearest the tip) of your little finger which is curled up.