Like most things, it is ultimately more complicated than that, but I think there is some validity to the theory. If Octavian had died a couple of years after Actium, the cycle of civil violence almost certainly would have returned. I would argue that the machinery of the Republic was permanently broken by this point, and that the most obvious culprit was Pompey (but that's another topic). By living a long time (relatively speaking) Octavian retained power long enough to consolidate and normalize the new order of the principate.

I would argue that the principate/monarchy/whatever didn't really start with Augustus, but really officially began with the transition of power to Tiberius. That bloodless transition was an important herald of the new hereditary system. It wouldn't always be so easy (like in 69 CE, for instance), but the precedent was set. The long reign of Augustus was a sort of transitional period, a bridge between the Republic and the Empire. If you look at some of the events during his reign, you can see this in action.

First of all, Octavian was savvy in representing himself as a 'normal' Republican magistrate. By his time, 'normal' meant something beyond simply being a consul, but as far as the precedent set by Sulla, Pompey, and Caesar, he followed along. He held extraordinary imperium like them; he was a triumvir "for the putting the Republic back in order" like them; he completed the public building projects of Julius Caesar (the temple of Venus Genetrix, for instance); and he fought wars for the benefit of the SPQR, like them (even if Agrippa actually did all the heavy lifting).

After Actium, he began a very clever program of easing the state into something new. First, some appeasement: he nullified all the acts of the triumviral period that were contrary to the mos maiorum (popular custom); he re-instituted election by the assemblies; he completed the first census of the Roman people in 42 years. In 27 BCE, he and the Senate met to assign provincial governors to their provinces, just as it had always been done before the triumvirs took over. The Senate was only too happy to ensure that Octavian received extraordinary command over the "dangerous" provinces (ie the ones with large standing armies in them), setting the stage for the institutionalized "Senatorial" and "Imperial" provinces. Areas with opportunity of further conquest were quietly assigned to members of the Imperial family, and that somehow was okay, even though there had not been that kind of family nepotism in Roman politics for centuries. These family members (Germanicus, for instance) were, officially, Roman "proconsuls" invested with power by the SPQR, and everything was above board.

But after and during this appeasement, Augustus was making subtle changes to the basic structure of government. After 23 BCE, there were no more annual consulships for Augustus. Essentially, he abandoned the basic symbol of Roman power because he didn't need it anymore. The entire character of Rome's chief magistracy changed drastically, permanently. Augustus himself took on instead a permanent "tribunician power" (which I won't go into, but it was powerful in a subtle way and would be a cornerstone of the Emperor's basis later). This was very anti-Republic -- it was an innovation.

At certain points, he had to contend with potential rivals, and he was careful to do so. There would be no new civil war, and no one would be allowed to amass power enough to challenge the Augustan stranglehold on legitimate force. The first governor of Egypt as a Province, Gallus, made the mistake of bragging too much about his exploits on the Upper Nile. There was no longer any room for such boasting, and he was dealt with. M. Primus had gotten a little adventurous with an "unauthorized" war in Macedonia, and he too was dealt with (23-2 BCE). There would be no more Julius Caesars, conquering (or threatening to conquer) great swaths of territory and raising multiple legions in the process. Egnatius Rufus, in 19 BCE, dared to stand for the consulship without the expressed approval of Augustus. This too, was no longer permissible. Candidates for the consulships were now gently suggested by Augustus (read: appointed).

All of these changes came gradually, and at the hands of a man beloved by both the people and the Senate. The people were happy to finally be rid of decades of civil war, upheaval, privation. What was left of the Senate was happy to end the cycle of powerful principes vying for ultimate, extra-Republican power -- at least that is how their behavior towards Augustus reads (including giving him the title "Augustus"). The fact that he lived so long allowed Augustus to mold things slowly, and allowed him to build up auctoritas (non-specific political authority) needed to make the changes without raising many eyebrows. So in a sense yes, I think his long rule was important for the transition. That was rather rambling -- maybe some one else will chime in too.