One thing I've noticed is that they seem to be fairly weak at close range, as long as you don't go all the way into melee. At this range, she won't charge around or fire her projectile attack. Instead, she will just walk slowly towards you and occasionally use her area blast.
The strategy I've been using so far with my Vanguard on Insanity is to first try to isolate the Banshee, and then keep doing Charge, quickly step back out of melee range to avoid her instant kill attack, then keep shooting from close range until Charge is ready again.
While it's fairly easy to see when she's about to do her area blast, I usually ignore it. It has so far only done enough damage to drop my barrier, which I can quickly get back up again with another Charge.
Make sure you have some room behind you to maneuver so you don't end up trapped in a corner, and she should die after a few rotations.
During the later levels I found Liara and Kaidan's Warp + Reave combo to be extremely effective for dropping armored targets like Banshees, Brutes and Ravagers quickly. Combined with Energy Drain as Shepard's bonus power, that squad setup works really well against Reaper forces in general, as it has plenty of powers for dealing Armor/Shields/Barriers, lots of ways to set up biotic combos, plus Liara's Singularity and Stasis for crowd control.