I'm going to start this answer by quoting a famous passage from the introduction to Tacitus' Annals - he says it far better than anyone else can:

"When the killing of Brutus and Cassius had disarmed the Republic; when Pompey had been crushed in Sicily, and, with Lepidus thrown aside and Antony slain, even the Julian party was leaderless but for the Caesar... Opposition there was none: the boldest spirits had succumbed on stricken fields or by proscription-lists; while the rest of the nobility found a cheerful acceptance of slavery the smoothest road to wealth and office, and, as they had thriven on revolution, stood now for the new order and safety in preference to the old order and adventure."

Augustus' victory at Actium and sole grasp of power from 27 BC was the final act in a series of civil wars that had last for almost two decades. Tacitus lists the crucial moments, but we must remember that it wasn't just the big names that died but often large numbers of their supporters. Twenty years of open fighting, political proscription and natural wastage meant that the senatorial order was nothing like it was when Caesar was assassinated, let alone the decades before (Tacitus makes exactly this point a little further on). All those who died in those years were the resistance to Augustus. Anyone left by 27 was either terrified, uninterested or on the side of the victor.

That's not to say that Augustus was totally free from danger, but the nature of the evidence makes it incredibly difficult to see in any detail. Suetonius (Augustus 19) says that any attempts at revolution or conspiracy were stamped out before they became dangerous to the Princeps, but nevertheless goes on to list nine seemingly separate ringleaders of various plots. Whether it counts as political opposition or not I don't know, but there's certainly something fishy going on with the list of men accused of 'depravity' with Julia, Augustus' daughter (Vellius Paterculus 2.100): Quintius Crispinus, Appius Claudius, Sempronius Gracchus, Cornelius Scipio, not to mention Iulus Antonius, son of the famous Mark Antony. These are some big names (and the last in some long and illustrious Republican families), so I'd be surprised if there wasn't something more than a bit of illicit shagging going on.

For the most part though, Augustus' supreme auctoritas, coupled with his total dominance of the military and his massive popularity with the urban poor would have kept the remnants of the old aristocracy in line. It seems that anyone who crossed that boundary was quite quickly dealt with.