I agree with BOSdev that you can find an apartment closer to your price range. Since I imagine you will be traveling by car a lot for your job, I would also consider living near a freeway. If you are young and want a big city experience you might consider giving up some space and paying more to live in Cambridge or the Backbay, or a loft apartment DT or near the green line D branch or Beacon St Branch in Brookline or Newton All of those areas are very safe. Many young people are moving to south Boston which is near DT but has waterfront urban beach propoerty. No part of Boston is really dnagerous compared to some big cities but you might avoid Roxbury and Dorchester as probably the most blighted areas of the city. Everet, Revere and Charlestown can be difficult to get in and out of by car. I would suggest you pick some places on the internet and then visit the city for a few days. There are very cheap flights into Logan, Providence or Manchester. It's very different from where you live now, with older houing stock and much denser living.
To prove you need more money to live in Boston do rent and salary comparisons on the internet and show them to your boss.