Cops accepted him. He could freely take parts in the investigations and openly cooperated wth Gordon. During his first appearance there might have been mixed opinions because he was knew and people didnt know much about him, but between the movies he saved the city quite many times. We see people dressed as him fighting crime, saying that hes an inspiration, so people know hes been cleaning up the city and nobody denied that when Dent said so during the press conference. After a while people wanted him to turn in because they were scared and wanted him to take the bullet for them. Yes, we only know about Gordon;s kid but thats also the only kid we get to know. And just look at the reaction Dent is getting while getting into the SWAT car when people think that he's Batman. Batman was no menace and even in the same movie its shown that he inspires people and that they know he fights crime. While I see why they didnt want to reveal Dent's insanity (although taking the circumstances into consideration, it couldve been explained shock or brain damage after the VERY recent accident), the blame could have been put on just about any one of Joker's people in Batman's outfit, and that would save all of Gotham's symbols - both the white and the dark knight