Not having a car is going to restrict your options outside of New Haven severely, unfortunately. I am not familiar enough with the public transit system to help you on where you could look outside of New Haven.

As far as websites - I would avoid Craigslist entirely. Zillow, Trulia, and apartments.com might be worth checking out, too. If you're looking for roommates, though, the Yale off-campus Facebook group will probably be your best option. You (likely) won't find people looking for roommates on Zillow or Trulia.

Anything asking for a deposit and banking information up-front is 100% a scam. Do not engage with those listings whatsoever.

Affordable housing will be much easier for you in New Haven if you can find roommates. Otherwise, you might want to look into neighboring towns. West Haven and East Haven are probably your cheapest neighboring options. Hamden, North Haven, and Branford are not quite as cheap as those last two, but are nicer than West and East Haven (avoid the part of Hamden close to New Haven... I would restrict your Hamden search to zip code 06518). You could also look at Wallingford for dirt-cheap electricity rates, but it's the farthest drive of the places I just listed.