Branford Ct Apartments For Rent . The average home rent in this city is $2,989. Try rentals.com to compare amenities, photos, & prices to find apartments that match your needs. 214 Opening Hill Rd Branford, CT, 06405 Apartments for Rent Zillow from www.zillow.com See all 58 apartments currently available for rent. View houses for rent in branford, ct. Facilities include a small playground and picnic area.