Hard Drugs Never Hit the Streets
The beginning of the end for the mafia was when drugs came involved in mafia business. The mafia made a pretty tidy racket doing some rather laughably mundane things prior to the hard drug trade. One such racket was stealing tanker trucks full of gas and using them to refill mafia owned gas stations. The gas stations would sell the gasoline without having to pay the taxes or purchase the gasoline. Rackets like this would bring in anywhere from \$15,000-\$20,000 a month. That's good money, but it's not "beat somebody to death with a baseball bat and ditch their corpse in the harbor" levels of income. People usually don't start committing heinous impossible to ignore crimes like torture followed by murder and mutilation over numbers like that.
Once drugs became the mainstay of mafia income you're talking about street level goons suddenly running drug rings that are cranking out a million and a half a month. The mafia wasn't ever really comprised of sophisticated people, they were largely uneducated and their scope of reach wasn't really all that great. They tended to for the most part be dumb thugs who might rough you up but rarely killed and usually had to watch their backs because johnny law would get interested in too many corpses started turning up. Once the mafia really became too successful due to drugs is when you saw massive infighting and a lot higher level of violence on the streets. This ties in with something else as well.
The War on Drugs Never Happened
Today we simply take it for-granted that entire swathes of the city are junkie riddled virtual no-go zones. It really wasn't always the case. When honest to god hard drugs like crack and heroin started hitting the streets it hit the lowest common denominator of every city like a figurative nuke. Massive crime waves, massive social decay, it terrified people. Street level meat heads could suddenly make enough to switch out their crappy .32 autos and .38 special revolvers for Uzis and street sweeper shotguns. Guys with switch blades who would have been content to just stick you up for your wallet were suddenly hooked on crack and shooting up the corner store with 1911's and Berettas to get their money for their fix. Lawlessness had always existed on some level, but hard core drugs hitting the streets changed things pretty radically. The war on drugs was a knee jerk reaction to this wave of violence and crime that was threatening to overtake our cities.
One of the biggest death knells to the Mafia rose out of the War on Drugs. Mandatory Minimum Sentencing. Believe it or not there was a time in our history not too long ago when getting caught committing various felonies only netted you 3 to 4 years in the clink. You could get caught with a bunch of stolen goods, or illegal weapons, or hard drugs, or even murder and not actually get all that much time in prison for it. The new much harsher sentences created by Mandatory Minimum Sentencing meant that once a made man was nabbed by the cops he might be taken off the streets for decades. This really harmed to mafia's ability to maintain and pass on power. Previously getting arrested was just an inconvenience, albeit a major one. Now it was a career ending event.
What's more, when the big Dons started getting locked up by these new harsher sentences the guys who replaced them were not very smart. The real dons were replaced by street thugs in fancy suits. When you give a guy whose job qualifications are "breaks kneecaps real good" and "scares store owners into paying their protection payment without a fight" several million dollars and the reins to a few hundred guys just like him things get... messy. Once the officers and bosses got locked up and the soldiers started running things is when the crazy over the top violence and endless cycle of back stabbings happened. You can't run an organized criminal enterprise that violently and get away with it for long. Thanks to mandatory sentencing decapitating their actually qualified leadership they started killing each other over the drug money they were making like feral dogs fighting over a steak, or got arrested for multi decade long sentences.
This Means That The Police Never Militarized
Again, up until very recently in police history the cops weren't really very well armed. They didn't need to be. A .38 special K-frame revolver is more than enough to handle some punk with a switchblade or a .25 auto pocket pistol. Today, modern police are basically a military organization complete with huge budgets, machine guns, hard core organization, and nation wide data (intelligence) sharing and collecting. This largely happened in response to the crime waves sparked by hard drugs hitting our streets, which largely happened because the mafia realized they could really make a ton of money selling the hard stuff and started the first major drug smuggling operations on our shores.
More devastating to the Mafia than cops with bigger guns, was the addition of military level surveillance tools to the police's arsenal. The various espionage gadgets that police got a hold of really were nothing the mafia had ever seen before. They knew about wiretapping, but had no idea about remote surveillance like hidden cameras, bugs, worn wires etc etc. They really had no idea how to deal with it. Quite a few mafia higher ups got blindsided by the police using tools that had formerly been only available to the CIA. In addition to this, departments began connecting more and more. Moving around wasn't protection anymore since with the click of a mouse every last detail down to your fingerprints could be available to every cop in the USA. These tools introduced to the police alongside the rest of the revamp the police got as part of the war on drugs really tore through the mafia something fierce.
I think that if somehow the hard drug pandemic that swept our cities never happened and the mafia never got in on it they would till be stealing gasoline tankers and shaking down business owners for protection money. The drugs made the mafia too much money, which made them too aggressive and too high visibility, which made them rather deserving scapegoats for the pandemic levels of street violence and crime that resulted from their drug trade. No hard drugs hitting the streets, no massive crime wave, and no draconian new sentencing laws and militant police to deal with. In addition to this, the 4 decade long morality crusade of "tough on crime!!!" wouldn't have placed the mafia directly in the cross-hairs as the (mostly rightful) scapegoats for the chaos overtaking the cities. The mafia see the hard drugs coming into their neighborhoods, but in this alternate timeline they resist the urges of greed and decide that crack and heroin cause too much damage to their community. Obviously if this stuff gets out of hand the police are going to go ballistic and become VERY bad for business. The mafia begin very strongly opposing the influx of hard drugs and a quiet and unofficial alliance develops between them and the cops. They keep selling TV's and VCR's that "fell off the back of the truck" and the cops don't notice as long as they keep putting the cops onto incoming drug shipments. The massive crime waves of the 80's never develop as hard dope remains a rather fringe thing that is hard to get a hold of (and VERY bad for your health to be caught selling.)