While the reason Publius Cimber has been banished is never stated in the play and remains ambiguous, some suppositions can be made.
Publius Cimber, a senator, appears in both Act II and Act III of Julius Caesar. In Act II, Scene 2, he enters with Brutus, Ligarius, Metellus, Casca, Trebonius, and Cinna as Caesar tries to dispel Calpurnia's dread that he go to the Senate. The presence of Publius may serve the conspirators as a decoy since he has a banishment upon him, and his pardoned brother, Ligarius, who accompanies him, may distract Caesar from the conspirators if he speaks to Caesar on behalf of Publius. Oddly enough, Caesar greets Publius warmly after Publius wishes him a good morning: "Welcome, Publius" (2.2.114).
Then, in Act III, soon after Caesar enters the Senate, Cassius worries that Publius may know what he and Brutus and the others plan to do because Publius says, "I wish today your enterprise may thrive" (3.1.14). But, right before the assassins attack Caesar, Ligarius asks Caesar to pardon his brother Publius and remove his banishment. Since Ligarius had supported Pompey during the civil war, it would seem plausible that his brother Publius had done so, as well, although nothing is stated. However, Caesar refuses to pardon Publius because he was sentenced under lawful decree. Taking advantage of Caesar's being distracted, Cassius and Brutus and the others stab Caesar, killing him. Afterwards, Publius is exploited by Brutus to inform the Roman citizens that aside from Caesar, no one else will be harmed. (3.1.96-98). Since Publius is out of favor with Caesar, he would have no reason to falsify anything, so the Romans should believe him.
So, while Publius plays a significant role, the reason for his banishment is never clearly established. Only the supposition that he is connected with his brother Ligarius in the intentions to side with Pompey against Caesar can be made. Even this is somewhat dubious since Ligarius has been pardoned, but Publius has had a lawful decree passed against him. Could it possibly be that Publius has tried an assassination attempt on Caesar himself in the past? Is this then, perhaps, why he has told Cassius, "I wish today your enterprise may thrive" (3.1.14)?