First, let's take a look at the actual set on stage. Most of the action takes place in the apartment of the Carbones. Stage directions tell us that, while it's pretty bare, it's still clean and homey. You can tell a lot about people by what their house looks like. This apartment would seem to say that these folks don't have a lot, but they take pride in the things they do own. Also, on stage there is a bit of the street outside. This becomes very important when all the family drama going on in the apartment spills out into the public arena. It allows for a space where we can see the community affected by Eddie's mistakes, such as when the Italian's are arrested or when Eddie and Marco duel.
In his very first monologue, Alfieri gives us a broader outlook on our setting. He says, "this is Red Hook […] This is the slum that faces the bay on the seaward side of Brooklyn Bridge. This is the gullet of New York swallowing the tonnage of the world" (1.1). We're in a hard core neighborhood, where people work their butts off everyday on the waterfront. People don't have a lot of creature comforts here. They have their families and they have work, hard work.
It's important to note that the characters in the play don't see this as a bad thing. To Eddie, it's a duty. He finds honor in his life as a longshoreman. To Marco and Rodolpho, the Italian immigrants, America as a whole represents work. It's a place of opportunity, where they can make enough money to be the men they want to be.
Also, let us not forget that this is specifically an Italian American community. The characters' ideas of respect and justice are based on their heritage. This becomes really important toward the end of the play, where "civilized" American law is totally dissed. Italian ideas of honor and revenge are what drive Eddie and Marco into their destructive duel.
The time period of the play is also important. It's the 1950s. The most telling sign is probably the treatment of women. Just flip on any old sitcom from the time and you'll see what we mean. Back then, the overall attitude was that men ruled the family. The women were supposed to do what they were told, stay home, and take care of the kids. The idea that a woman might have bigger aspirations than being a housewife was absurd.
In a way, Eddie is almost progressive by encouraging Catherine to go to school at all. Of course, he throws a fit when her schooling actually leads to a job. Also, it seems pretty obvious that Catherine's secretary job is viewed as a way to get a better husband. Once again, women are seen as needing a man to exist. The way Eddie relates to his wife, Beatrice, is also colored by '50s perceptions. Even though she speaks her mind, she still ends up doing whatever Eddie says. It doesn't matter if she agrees with him or not. He's the man.
We'd be remiss if we didn't mention how 1950s ideas of masculinity also affect the play. Rodolpho's supposedly feminine traits, like singing and cooking, wouldn't as big of a deal these days. In '50s Red Hook, however, Rodolpho is seen as strange wherever he goes. All the men down at the docks make fun of him. Eddie shares and is influenced by this perception of manliness. These ideas definitely fuel his suspicion of Rodolpho.