Shakespeare's Julius Caesar is a depiction of the downfall first of Julius Caesar, and then of the conspirators who assassinated him, including Cassius and Brutus. Mark Antony, however, does not actually experience a downfall in this particular play. Indeed, he uses his considerable rhetorical skills to both engineer and end up on the winning side of the later conflict with the conspirators. As a matter of fact, at the end of the play he seems poised to acquire more power.
That said, Antony does experience a downfall; it's just not in Julius Caesar. Instead, Antony's downfall occurs in another play, Antony and Cleopatra, which depicts events after the assassination of Caesar and the defeat of Cassius and Brutus. In the play, Antony is a member of a triumvirate (three men ruling the Roman world). Many things contribute to Antony's downfall, but one of the most important is his obsession with Cleopatra. Antony's infatuation with Cleopatra causes him to become blind to the tenuous nature of his situation, and also to neglect his duties as a leader. For instance, when Cleopatra's ship flees the battle of Actium, Antony follows suit, leaving his army to suffer defeat at the hands of Octavius, Antony's former ally in Julius Caesar and the future emperor of Rome. As such, though his obsession with Cleopatra is not the only quality that leads to Antony's downfall, it's certainly one of the most important.