This site's helpful:

https://www.padmapper.com/apartments/stamford-ct/1-beds/under-2000?box=-73.59244,41.0149,-73.49594,41.10567

As people mentioned, watch out for hidden fees (parking, pets, >1 month deposit, etc.), but it's not crazy to think that you could find a decent place for <$2k before paying utilities and such.