You may have some luck in Allston/Brighton. I assume you want a 1 bed - it won’t be impossible to find one at $2K or less but these get snatched up pretty fast. A one bedroom that’s $2k or less, with no broker fee, and close to the T is going to be harder to find and even harder to secure because a lot of people will also want that great of a deal.

You will likely need to increase your budget by $100-200 and compromise on the broker fee. Maybe you’ll get lucky but I want to be realistic with you. I still think Allston/Brighton is your best bet especially if you’re concerned with safety and still want to be in city limits.