So many ways to play an open G. Have you tried playing it with middle, ring, and pinky instead. Also, as DavesGuitarJourney points out, the difference betweern playing it with the 5th string B or without is subtle but arguably cleaner sounding without. To do that, you can ignore your index finger (I lift it a little bit) and deliberately let your middle finger drape over and mute the 5th string. For that matter, if you really needed to get a G chord in, you could just play the bottom 4 strings with the high E fretted on the 3rd fret and the B, G, and D strings open (so the D would be on the bottom), or lift your middle finger off the 6th string and don't play it. All these versions sound pretty similar to me and they are all G chords.

To help get down the fingering you are having trouble with, maybe just repeatedly strum it while making small adjustments to your hand position, finger position, angle of fingers with respect to the fretboard, elbow/shoulder position as William suggests. Is there a hand position, maybe dependent on the positioning of the neck, where the accidental muting isn't so much of an issue? I play this chord with my thumb projecting over the fretboard a bit. In that position I find that a VERY slight increase in the angle my wrist makes with my forearm can make a big difference.So I'm pushing my wrist down, rotating my hand just a fraction which gets my fingers a bit more vertical over the fretboard.