Is transporting passengers—a taxi driver—to or from places like bars or brothels forbidden (haram)? And what is the Muslim's duty in this situation?

Transporting people to brothels or places of corruption is considered cooperation in sin and aggression, and one should avoid that. "And whoever fears Allah – He will make for him a way out." So, seek another job if your current one necessitates something forbidden. However, if you are unaware of their intention, then the wage is not unlawful.