My interpretation of whatever monster I feel like talking about, primarily from the 1st ed AD&D Monster Manual and Fiend Folio, but not limited to them, treated in a way that deals with the stuff I find interesting. That is: ecology and culture, if applicable. Stats can go hang.
Banshees are referred to in the Monster Manual as Groaning Spirits and are the spirits of evil elves that cause you damage by screaming at you. As it would. I’ve had my share of evil elves screaming at me in my lifetime. It’s very disturbing and psychologically scarring.
But that’s not a Banshee.
It is the tradition in many lands to hire mourners at the funeral of prominent citizens. This is a profession, a form of musical expression, and if you have to tie it to a category you could call them Bards of a certain kind. The very best can earn great sums of money. Because, you know, nothing more demonstrates to the peasants that you are really important, even in death, than some unknown woman shrieking to put your teeth on edge hired at great expense.
So much does God love the rich and powerful that sometimes He allows these mourners to continue their work after death. These are the real Banshees, paid for with an unknown and unearthly currency.
Now it is a funny thing about the Otherworld that time is not the same linear regular material that we think it. Sometimes it flows faster or slower there. Sometimes it flows backwards or zigs and zags, bubbling and boiling around unseen objects in the flow of reality, like water around rocks in rapids. As a consequence, sometimes a Banshee can appear to do her duty to the sadly deceased before that person is even dead yet.
This is of course a disturbing event for all concerned, and might just have the consequence of causing a heart attack or so traumatising the named person that they fade away and die. Making it a self fulfilling prophecy.
At other times a Banshee may appear to the family when the deceased has passed away far from home, making them a kind of cosmic telegraph service.
From a game point of view, a Banshee does not appear and attack you. That’s just silly. It is there to mourn for someone, in a dispassionate kind of way. Importantly, this actual or imminent death is a fact. If it has not already happened it is definitely going to, and soon. Unless you are planning to categorically kill off a player character, therefore, a Banshee should not appear to them and sing for them. They can witness it being dedicated to someone else, and this can be a great spur to adventure.
- Prominent citizen is visited by a Banshee. Clearly he/she is about to be assassinated and hires the party to protect him/her (the effort will inevitably be in vain, but it does lead on to the rest of a larger plot about who planned it and why)
- Prominent citizen is visited by a Banshee, and recognising that his Fate is about to catch up with him spills the beans on some great secret to the players
- A Banshee appears and keens for a great hero far away, alerting the party that he has died and that an expedition to recover his remains and/or possessions could be launched.
As a final thought, if you include Sanity rules, then I think the visitation of a Banshee would definitely cause psychological stress to the players, necessitating some kind of check. It is a concrete expression of the supernatural, after all, and one that challenges traditional notions of space and time.