If it didn't start the trend, The Godfather is one of the catalysts in the popular trend of glorifying Italian mobsters and gangsters before much of society did not think highly of these individuals/organizations.
Hollywood started the trend of these types of people/organizations becoming glorified, dignified, looked up to as honorable, and prestigious with a larger-than-life personality. Not to mention all these nicknames they go by, this is what started rappers mimicking that lifestyle and since then a lot have built their careers off of that infatuation.
From movies to TV shows, this is where the infatuation with mobsters, gangsters, druglords, etc began. The '70s, '80s, and '90s popularized these things with all those movies and TV shows being made about them, people took a liking to the characters and scenarios and started imitating them. The Sopranos brought it to its peak.