There has been three different football teams named after the Brooklyn Dodgers baseball club
The Brooklyn Dodgers who were part of the NFL from 1930-1943
The Brooklyn Dodgers who were part of the AAFC (All-American Football Conference) from 1946-1948
The Brooklyn Dodgers of the Continental Football League of 1966
NFL Brooklyn Dodgers from 1930-1943
As told by a few historical sports sites this team was highly associated with the Brooklyn Dodgers Baseball team. Like the NY Giants baseball and football teams, Brooklyn shared the name Dodgers for baseball and football. It was a much more profitable recognition and it was better for business. Both Brooklyn Dodgers team shared Ebbets field, as they would not overlap due to football being a winter sport.
The team played for 13 years and only had 2 winning seasons in which they landed in 2nd place. The franchise all time record was 60-100.
The Dodgers made NFL history on October 22, 1939. That day, at Ebbets Field, the Dodgers played the Philadelphia Eagles in the first NFL game shown on television. The Dodgers won the game 23–14.
Another interesting tidbit is that this franchise has indirect ties with the Indianapolis Colts, as the franchise folded in New York, the NFL moved the team to Texas and became the Texans. After another disaster team all the players were moved to an expansion team in Baltimore the now Indianapolis Colts.
AAFC Brooklyn Dodgers 1946-1948
The reason why the NFL Dodgers folded was because owner thought the AAFC would offer more incentives thus moved the franchise to All-American Football League. During the 1940's the AAFC had better players and was challenging the National Football League.
As for the AAFC Dodgers, they had three horrible seasons and decided fold and merge with the AAFC Yankees renaming the team to the Brooklyn-New York Yankees
Brooklyn Dodgers Continental Football League 1966
This team actually had no relations to the Dodgers baseball team, to the extent that the reason why the team only lasted for one season was because the Dodgers actually sued this team for copyright infringement.
The team tried to gain viewers and fans in Brooklyn as the baseball team was getting ready to move to Los Angeles that even hired and named Jackie Robinson as their general manager.