Brobbey was at the centre of a key incident when Tottenham captain Cristian Romero collided with his own goalkeeper Antonin Kinsky under pressure from the Sunderland forward.
Romero was forced off in tears and Kinsky completed the game with his head heavily bandaged.
"We stand firmly with Brian and offer him our full and unwavering support," Sunderland said in a statement.
The club added that the abuse was not an isolated case, saying Romaine Mundle and Lutsharel Geertruida had been subjected to similar treatment in recent weeks.
“The club has reported this incident to the Premier League, the relevant social media platforms, and the police, and we expect swift action to be taken against those responsible," they said.
Sunderland climbed to 10th in the table and will face fourth-placed Aston Villa on Sunday.