How do I stop that banshee from eating my face?

My strategy for the Banshee is fairly class agnostic.

First, try to wipe out everything else that you can, so that you're down to just the Banshees left. This can be tricky, but they tend to move more slowly at first than other enemies, so if you put some distance between you and them, the other enemies will rush you and you can focus your fire on them.

Mostly my strategy is to run. I run like hell. The Banshee's really only got a couple of attacks. First, don't let them get close to you, and second, keep enough distance that you can rapidly turn a corner if they throw their warp projectile at you. Then it's just a matter of wearing them down. The heavily armored, so use whatever anti-armor gear/powers you've got on them. Their barrier recharges relatively quickly, but it's pretty weak by comparison.

Getting hit by their attacks not only does a ton of damage, it also causes you to take some damage over time. Prioritize getting enough ground and cover between you and them to block their ranged attack when it comes. If they start teleporting, just start running.

I try to train the Banshees myself, and keep my squadmates alive and focusing fire on them one at a time. My squadmates really couldn't tank the Banshee worth a damn - they tended to go down quickly if it focused on them. If they go down, do a couple of laps of your arena to get some distance, and you should be able to spare enough time to get them back up again. Having maxed out medi-gel so you can just instant-revive them and/or recover your own health is a plus.

As an Infiltrator, I found dealing with these things a pain (in general, and to my ears, as that screeching gets old fast). However, I was able to find a combination that seemed to deal with them effectively enough, which may or may not work for you depending on your chosen character class.

First, I beefed up Warp Ammo, opting for the Damage-based options for power tiers four and six (for tier five, I went for the ammo bonus since I hate running out of thermal clips). With an appropriately upgraded rifle, this gave my sniper shots enough umph to make sizable dents in both the banshee's barrier and armour.

I then fully upgraded Incinerate, opting for the damage, burning damage, and armour damage options in tiers four through six, respectively. Since the banshee's base "health" is just armour, the 50% damage bonus offered by the last tier was particularly handy.

With powers taken care of, down on the battlefield it became a matter of alternating between sniper rifle rounds to the head and blasts of Incinerate. The timing of the shots was just long enough for Incinerate to recharge in my case, and so long as I was careful to reload the sniper rifle manually when necessary (due to the Incinerate interrupt), I could pretty seamlessly switch between the two.

In the event that the banshee got too close before I could bring it down, I simply let my squad members distract it while I relocated to new cover. Infiltrators get a little benefit from Tactical Cloak here too, since in addition to helping relocate relatively undetected, it can also increase sniper rifle damage.

Admittedly this strategy was a little problematic toward the very end of the game due to the sheer number of enemies, but in general it kept me alive without too much difficulty.

Shotguns are your friend.

They are difficult to hit at long range, but once up close they are fairly trivial (just don't get impaled).

Since they also act as a support caster, they will sometimes create a barrier for allies. Knocking down this barrier causes them to stand still and scream for a while. This is your chance for a quick kill as they won't do anything else.

Finally, when they are charging up their nova attack, they will just stand there. Of course, the nova is deadly if they actually get it off, try a sniper rifle.