Regional airlines are profitable. Regionals make money (a small amount though) otherwise, they wouldn't be in business. Some of the regional flights might cost the mainline carrier more money than the amount of revenue they get, but on a whole, they make money too.
While CRJ with only 20 passengers that paid $200 for a ticket might seem like an unprofitable flight, the regional that operated that flight got paid a fixed amount. Those 20 passengers then boarded United flights which charged them $1,500 to go to Europe. If United didn't subsidise that CRJ flight, those 20 passengers from DCA wouldn't have been flying to Europe on United.
Mainline carriers cover the costs of regional airlines in order to funnel passengers from small cities that cannot support mainline aircraft, to larger airports that can. While the individual flights may be unprofitable, the total journey for each passenger is profitable (most of the time).