I'm not sure if I understand your post, but you do not need to frame a new ceiling underneath the existing floor joists. Simply gluing and screwing the drywall to the ceiling joists (the floor joists of the 1st floor) should suffice. In some situations, 1x3 furring is attached to the floor joists with screws or cross-nailing to "level " the basement ceiling before putting up the drywall or to provide extra depth for recessed light cans, or to create chases for cables or pipes that run below the basement ceiling joists.