You can find basement apartments for rent on rental listing sites, such as Trovit.com, Craigslist.org and Rent.com. Short term rental sites, such as Airbnb.com, also commonly feature listings for basement rentals, with pricing on a per night or per week basis.
Many of the listings for basement rentals offered by Craigslist or Trovit are mixed into the general selection of apartments or rooms for rent, which means the best way to locate them is through a keyword search. Searching for a term, such as "basement apartment," on either site displays all listings that contain that phrase, or a portion of the phrase, in either the heading or body. You may also need to enter other search criteria, such as a price range or specific geographic location, to see the most relevant selection of listings.
Airbnb.com allows its users to rent out rooms or sections of their homes, including basements, to different guests. The site functions more like a hotel reservation site than a property rental site, which means you may not be able to view the basement in person prior to your stay. As the company does not endorse any of the users who use its site, always exercise caution when staying with someone you don't personally know.