I find it difficult to do octave jumps where one string is open and the other is fretted. Eg. an open E-string and a fretted A-string.
I mute the A-string with my left hand by simply not pressing the string against the fretboard but still touching the string. But I can’t seem to find a way to mute the open string which just keeps ringing.
Any tips? 
Depends on the tune I guess but I’d either slip my thumb down from the pickup to the open string or use the non currently plucking finger to mute the open string.
Either the thumb mute as already mentioned, or go up a string (and 5 frets).
I’ll echo @HighlandBass and say, when there’s a weird or difficult spot in a song (like this open string octave thing), I usually will tranpose…translate…uhm… move the notes up the board so I’m still hitting the same notes, just not open.
So yeah, do it on the 5th strings. Problem solved.
Of course, shortcuts like this is problem one of the reasons why my muting still sucks.
If it’s the E string you can mute it with your thumb or tap it with a finger on your right hand
If you play an open E then play an E on the 7th fret of the A string there are two ways to mute the E.
First, just move your thumb from the pickup down to rest on the E. Second would be to leave your thumb where it is and as you pluck the A string, let your finger follow through and come to rest on the bottom side of the E string.
Don’t try to mute with your fretting hand because that would be awkward and you’d have to leave a gap between your fingers and the A string so you don’t accidentally mute the A as well.
I agree, if you can semi float the thumb it makes muting a breeze.
I play with a pick. One way I would do it this: Let’s say I need to play E - A - E. I’d pluck down on the E and as my pick lands on the A string I reach over with my left hand and gently tap the vibrating E string. I would then pluck up to catch the A, mute it with my finger and continue upward to catch the E.
This sounds harder than it is. I work my left hand and right back back and forth like two kids on a teeter totter.
If you’re playing with fingers it’s easier to silence the strings right after you pluck.
What song is this? Mind sharing the section?
Octave usually a teetering from one root note to another so muting usually happens automatically. Even the octave that is use as accent then there are plenty of opportunities to mute either end.
There’s another option to mute the open E which is the left thumb. Nice and quick touch anywhere on the string does the job. Doing the octave on open E your left thumb would already up and over so it should be relatively simple and easy. That’s my experience unless I hear your track. Maybe it’s more complicated than I imagined 