Interesting question. A clear answer doesn't immediately pop into my head, so let's look into it and see where we end up.
As I understand it, government legislation for such things is a very 20th-21st century thing and perhaps anachronistic to extend to the ancients (for whom public drunkenness was not only a fact of life but sometimes encouraged in certain contexts: see Carcopino 262). Ancient attitudes toward drinking were not the same as they are today. For instance, the future Emperor Tiberius, while a general under Augustus, had a reputation for drinking wine while 'on-duty' so-to-speak, yet as he did not exhibit signs of drunkenness (a sign of indiscipline, lack of self control, and therefore unmanliness) nobody thought this was a particularly bad thing. He even had the amusing nickname of Biberius Caldius Mero, a pun on his name of Tiberius Claudius Nero, which basically meant "drinker of hot wine." This did not appear to detract from his reputation as a man of virtus and disciplina.
Another point to consider is that the games were essential elements of Imperial control over the large and potentially unruly population of Rome (231-3). At the height of the Principate, "there were 150,000 complete idlers supported by the generosity of...public assistance" and "perhaps an equal number of workers who from one year's end to the other had no occupation after the hour of noon..." (231). The point of the games was to amuse these people, and so I would imagine drinking would be a part of that enjoyment for some people.
On the other hand, the Roman elite had a poor view of the poor in general, and were particularly concerned about their congregation in bars, drinking, dicing and fighting (Beard 465-6). Claudius apparently abolished taverns, and banned the serving of boiled meat and hot water, the latter obviously being to mix with wine (but as Beard says, he didn't ban the wine? see p. 465). Even so, the Roman state had no means to enforce any such restrictions (456), and this can only have applied as well to the games at the Colosseum, where the Emperor (or in earlier periods, whatever magistrate responsible for the games) intended to bask in the adulation of the public thankful for the games he provided.
That said, the highly flavoured and watered down wine that the average Roman drunk, being purchased from taverns and bars, which might be kept in a skin for such an occasion as the day's games, is certainly not the same as taking a case or a couple bottles of JD to the Colosseum. While there was no Security at the door and the Praetorians wouldn't have been throwing people out for being drunk, the availability of alcohol in the modern age with shops, bottles/cans and disposable income was not equalled for the average poor Roman. The games were free, which is why the poor could go and enjoy it: wine wasn't free, and I doubt they all had the cash on hand to take a literal skin-full to the games and imbibe it, considering that games were on basically every other day. As an aside, these plebians would be more likely to get a seat in the Circus Maximus (250,000 seats) than the Colosseum (50,000) seats.
So basically, I think some people would have been drinking and some people would have been drunk and perhaps aggressive, I don't think it would have been on the same level as an English football game where everybody can afford and easily obtain a dozen-or-so beers during the course of the spectacle. I would suggest that drinking and/or drunkenness at the Colosseum would be less widespread than at a modern sporting event despite the lack of restrictions.
As for avoiding violence, well, there was no police force. But there was a strong military presence in Rome of the Principate, consisting of tens of thousands of troops, most prominently the Praetorian Guard. The elite's general concerns about the gathering, dicing and drinking of commoners in taverns was no doubt due in part to the fear of some sort of mob uprising. But while the games were there to entertain people, and to bind them in a sort of mob loyalty to the Emperor who provided them these things, if things turned ugly then the crowd would come off second best. Perhaps the most prominent example of violence breaking out at such an event is the Nika Riots in AD 532 (admittedly in a later period and in Constantinople), which nearly toppled the Emperor: but in the end, he sent in the troops, and tens of thousands of rioters were killed. The grain dole and the games were a very calculated institution that bound the Roman people to the Emperor and gave them plenty of reasons not to riot against the person who provided them all these things, but if that relationship broke down, the Emperor had thousands of professional soldiers on hand who were more than willing to massacre those who opposed them. These troops would also have played a more small scale role in controlling violence at such major public events in order to nip disorder in the bud, as the Emperor had such a vested interest in these events, but the exact details of their involvement I cannot say
Sources:
Beard, SPQR, 2015.
Carcopino, Daily Life in Ancient Rome, 1941.