Not a villain at the moment, Kate is her carer now - since even places that aren't Arkham Asylum, in this universe, can't keep their patients safe from abduction/escape. So the sisters live together, and apparently Beth isn't causing problems at this time, and is on medication.
Beth/Alice was the following under each writer:
Rucka = Villain
J.H. Willaims III = Damsel
Andreyko = Anti-hero
Bennet = Damsel
Which leads us to where the character is currently Kate's ward, but we did see Alice in a non-cannon story a little while ago where she made IMO a great villain.