Muting is not something that comes naturally, it has to be worked on until it becomes second nature. And you need to learn to do it with both hands. Best advice is to get a teacher, but I will try to put some basics into words.
Playing primarily five and six string basses, I have even more strings to mute than you do.
When playing on the higher strings, I use the 'floating thumb' technique. For example, if I am playing on the D string, my thumb on my right(plucking) hand is against the A string, and laying across the B and E strings. So this allows me to mute those three strings. If my thumb were planted on a thumbrest or the pickup, I would have to try to mute those strings some other way.
When playing on the lower strings, I lay whatever finger(s) that are not being used to pluck a string across the higher strings. So, if I am playing on the E string, my pinkie is laying across the A, D, G(and sometimes C) string.
You also need to put your left hand to work muting, too. When my index finger isn't being used, it is deadening strings that are not being played on. Same with the other 3 fingers. This takes more work than right hand muting, but once you learn it, it can really help clean up your technique.
When I was learning to slap, I used to see people slap and wonder how they hit the right string every single time. Then one day, I watched one of my friends do it close up, and realized that he was often hitting more than one string, but only the one that he was playing on was ringing out. I looked more closely at his left hand, and the fingers that weren't fretting a note were muting the strings on either side of the one being played.
I told him 'Man, you really have that muting down!' He was like 'What?' and I pointed it out to him.
He said that he had been doing it so long, that it had become second nature, and he didn't even think about it, or realize he was doing it any more.
That is the key. And the only thing that will get you to that point is practice.
Hope this helps!
