it depends on what im playing. if its heavy and theres a chance of me hitting the higher strings, i do with my fretting hand. otherwise i let them be open just for a little bit of vibration that they pick up from the other strings and it gives my bass a fuller sound.
I would say the plucking hand. Coz if u use your ring finger, you can then change the release time(time that the note rings) of the muted notes by moving ur finger closer or further away from the bridge. You still have access to all ur other strings by muting with ur plucking hand. Keep on jammin!!