If you're looking for cheap and close by, look at apartments in East Haven or Hamden, New Haven is a small city so even renting in a neighbouring town will allow you to get to work in around 15 minutes. I make the same amount as you and I spend around 1100$ a month to rent in Hamden. Apartments in New Haven proper can be pretty pricy because of the Yalies.

I lived in East Haven my whole life growing up, my parents still live there, as do most of my best friends, and I went through East Haven Public Schools. East Haven is HEAVILY Italian-American but it's changed a lot in my lifetime. Many of my friends at East Haven High came from many different backgrounds, Ecuadorian, Puerto Rican, Cambodian, you name it. It's a great place to raise a family, people are very friendly and the school system isn't fantastic, but it's pretty good for the kids that really want to do well. East Haven has really good food (John & Maria's, Twin Pines, and Gutiz Bakery) and it's not snobby like other towns on the shoreline can be. Obviously there is some crime, it's a pretty blue-collar town, but otherwise it's very good for raising a family. Most of the people in the area look down on East Haven because of it's working class vibe, but honestly it's a pretty friendly and diverse place, and honestly it's a much better bang for your buck compared to other New Haven area towns.