They say you shouldn't spend more than 30% of your income on housing but that's largely outdated and most people spend 50%. In a comment you said you're earning $85k, so your take home is roughly $4,700 a month. You can afford up to $2,300 a month for an apartment — especially if it includes utilities or parking or other amenities as many do downtown. You can also choose to live outside of downtown and get a spare room in a multifamily home somewhere and it'll probably be cheaper by $200 - $500 a month.

Stamford has better and worse areas but it's not like other cities where there are "bad" areas. As a general rule, anything west of the river of east of the train tracks is going to be a worse area (the "west side" and "east side"). This is also true for the South End for any building that isn't a BLT building ("Harbor Point"). Again though, these areas just haven't had significant investment in decades but they're not crime infested. Everything North of the downtown is standard suburbia and it gets less and less dense as you go further north.