Appearance and Behavior[]
The Banshee appears as a flying spirit of a woman with a black dress. Since they always giggle and scream while attacking and moving around, it's easy to guess where they have spawned without even seeing them. Her movement pattern is more or less random, but they always move only short ways while trying to get closer to you without moving directly towards you (like Lost Souls do), making it harder to shoot them.
They are however quite easy to kill since they move in straight lines, making it easier to see where they plan to go once they turn around. Also they try to keep their distance to you when you get to close to them, always trying to keep the same distance to you.
Your distance towards them controls how directly they fly towards/away from you. If you are too close/too far, they move almost in a straight line away/towards from you. If they are in perfect range, they move sideways.
Attack Pattern[]
Banshees stop mid-air before screaming, shoot about 5 green projectiles with high speed towards you in a pattern similar to the Frozen Soul Spell, so being to close to them while they shoot makes it near impossible to dodge it, so long range attacks are recommended.
Their projectiles do not move through walls, but if she hovers inside one, some of them might spawn in front of the wall, making their way towards you.
Hints[]
Long range attacks might be the best way to deal with them, especially if you figured out their movement pattern, however you will need a weapon with fast projectiles.
If that's not possible, wait for them to shoot, sprint towards them and use burst weapons. Any weapon which allows a 'Hit and Run' tactic like that should work (alternative fire of Hellish Ember would be the best).
Try not to get surrounded by them, since they can fly and will not block each other when they surround you, unlike carrots do, giving all of them the chance to attack you at the same time.