All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and messenger.
A person should try to change any evil deed that takes place anywhere, whether in barbershops or any other place if he is able to do so. The Prophet, sallallaahu ‘alayhi wa sallam, ordered the one who sees an evil to change it physically with his hand if he was able to do so. If he could not, he should change it by his tongue (by advising and/or condemning), and finally by (detesting it with) his heart, and he, sallallaahu ‘alayhi wa sallam, informed that this is the weakest level of faith. [Muslim]
If you were unable to change the evil, then it is not permissible for you to go to that barber as long as there are other barbershops. However, if you do not find any other barbershop and you are in need for that, it would be acceptable to go there.
Allaah Knows best.