There are some people who say that it is not actually the movie industry that has ties in the mafia. Rather, it is the actors and actresses who play certain roles that may be tied to the mafia in different ways. Do take note that most of the people involved in movies are only rumored to be part of the mafia.
There are some movies that may be connected to the mafia based on the storyline and the things that should be done. A few decades back, people were so much interested in stories about gangster wars. Hollywood had to produce those films because they know that these films would sell. Profit is always important in the movie industry.
The short answer to this question is that no one knows for sure, but in many individual stars careers it is strongly expected to play an influence. For an example, when Al Pacino played in the Godfather it was clear that some strings had been pulled to allow him to star in the infamous role. It is vital to note that these relationships with the stars and expected mafia members or organizations are all alleged.
A popular name that comes up is Frank Sinatra. His father was a member of a gang and was convicted of murder after he was found driving the car from an armed robbery. A surprising name is Tony Bennett. He is reported to have some ties to a family in the mafia and allegedly had a run in with a mafia hit man in the 1970s.