Do you believe Brutus is responsible for his own downfall in Julius Caesar?

Indirectly, yes, I do believe that Brutus is responsible for his own downfall in Julius Caesar. While it's Mark Antony who incites the Roman mob to turn against the conspirators, thus starting the civil war that results in Brutus' death, Brutus sets the stage for this catastrophe to happen.
Since he's a noble, honorable man who acts out of a selfless interest to protect others (specifically the Roman populace), Brutus rather naively expects other people to do the same. However, Rome is a cutthroat political stage, and Brutus should have expected his political companions to exploit his own nobility. By allowing Antony to speak at Caesar's funeral, Brutus unwittingly gives Antony the chance to manipulate the masses of Rome against the conspirators. By doing so, Antony gains a lot of power for himself, but he also ensures that Brutus and his companions are doomed. As such, though it's Antony who actually commits the acts that lead to Brutus' downfall, Brutus naively sets the stage for this to happen, and so he indirectly brings about his own demise. More than anything else, it's this detail that makes Brutus a tragic hero.