It's also stated as "once a bum, always a bum." My first citations are from the spring of 1934.
17 March 1934, New York Times, pg. 20:
Once a Brooklyn player, always a Brooklyn player.
17 April 1934, Newark (OH) Advocate, pg. 12, col. 1:
There is a saying among the camp followers - "Once a Dodger always a Dodger" - but there now seems no reason to abandon all hope in Flatbush.
19 June 1938, Washington Post, pg. X2:
Once a Dodger always a Dodger is baseball's famous saying but today that proverb is not ringing true to Burleigh Grimes.
13 March 1937, Lincoln (NE) Evening State Journal, pg. 5, col. 6:
NEW RULES FOR THAT
BROOKLYN DAFFY CLAN
Burleigh Grimes Does Not
Believe Once a Dodger
Always a Dodger
25 September 1937, Sheboygan (WI) Press, pg. 14, col. 2:
Once a Dodger, always a Dodger - as much a part of baseball as "strike three, yer out" - works in reverse, and the present Gowanus groaners, from Grimes down to the humblest ripe-fruit throwing fan, still nursess a grudge.
22 July 1938, Sheboygan (WI) Press, pg. 12, col. 6:
There is an old saying that "once a Dodger, always a Dodger," but when Manager Burleigh Grimes took over Brooklyn managership at the start of the 1937 season, he said "nuts. I was a Dodger once and they never hooked me up with all that nonsense. The daffiness boys are dead. Acrobatics, clowning, playing and loafing are strictly out."
13 February 1939, Los Angeles Times, pg. A11:
"Why not," said Hoyt, "once a Yankee, always a Yankee."
(New York Public Library CATNYP record)
Title: Once a Bum, always a Dodger / Don Drysdale with Bob Verdi.
Imprint: New York : St. Martin's Press, c1990.