When playing a fretted note on the A string, the fingers of your left hand have to pass over the D and G strings anyway. If you want to mute the strings you aren't playing, you can use this to your advantage. Allow your fingers to gently touch these strings, not enough to fret, simply to rest there.
Also, if you're playing an open note on the A string, you may wish to have the fingers of your left hand rest over the D and G strings (since they aren't up to much at that time anyway).
I often do this instinctually, and have seen many other bass players do the same.
(This will depend specifically on what you're playing, and in certain situations the right hand may well come into it. Also, as with many things, I wouldn't be surprised in another player comes in here with another technique that works for them. This is a good rule in general though imo)
My comment got too long, so in light of your further question:
Generally, muting the string below (as in closer to the E string) is achieved in the right hand.
E.g. if I was playing this https://youtu.be/pcawnRIyeok?t=180 , when plucking that first open A string, my fingers would land on the E string (and just sit there, not pluck it).
When playing with rest strokes, I personally tend to pretty much always have one finger on the string below the one I'm currently playing, and I alternate between my fingers for each note. If that makes sense. So if I was playing a bunch of notes on the D string in quick succession then I would alternate between my plucking with my first finger and second. The finger not currently being used would be resting on the A string, and the finger that just plucked the D string would immediately land on the A string to join its finger bro.
Play, and tell me if this makes sense. You'll be surprised how quickly things like this become second-nature after a while. See which techniques feel natural to you, experiment and eventually you'll get into a groove with it. And even if you get into that pattern, never be afraid to switch up and try something new if you want to :)