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.