Muting strings

What's the best way to develop technique for muting strings so they
don't ring out? I'm having trouble doing that with my left hand and
I've had to resort to trying to use my right hand to mute strings
instead. When I try to use my left hand fingers to mute strings. it
gets in the way of the notes I'm actually trying to play. Please help.

It should be a combination of both actually. The fretting hand mutes the
thinner strings. than the note played. and the plucking hand mutes the
thicker strings than the note you are playing.

Here is a SIMPLE adjustment that completely revolutionized my own playing.
I played the "R Thumb Anchor" way for about 18 years. I changed over to the
following when I started wrasslin` with the 6 string bass. There were two
more strings to mute! I play this way now on 4 string also.
Most bass players anchor their R thumb on the E string or
pickguard.INSTEAD: rest your R thumb directly on the Strings
You can easily mute all the strings you are not playing. Just put your
entire R hand on the strings. Your thumb will be WITH the other fingers. rather than anchoring separate from them. Your hand should
roughly look like it does when manipulating a computer mouse. Now stroke the
G string while the tthumb is laying gently on the other strings.
As you try the D. and E string --your R hand will naturally mute the
higher strings while fingering the lower notes.