Jewish Bader - and maybe Ader - families may descend from the family called in the middle ages A'ader-a-Ba-wa,
whom contemporane jews read as Adarbi.
They somehow may be related to persian Badrés as well.
Balázs Déri
In German language Bader was a profession that includes hairdresser, barber and leader of the public bath. It is still a very common name in Germany, Austria and Switzerland. In my own family tree I have more than 50 ancesters of this name.