Here is the entry for "in someone's corner" in Christine Ammer, The American Heritage Dictionary of Idioms, second edition (2013):
in someone's corner On someone's side, supporting someone. The term originated in boxing, where a fighter's trainer and assistant are in his or her corner (one of two diagonally opposite angles in a boxing ring) to provide advice between rounds. The figurative usage dates from the 1940s. [Example omitted.]
An early example of literal usage
Examples of literal use of "in [someone's] corner" appear in newspaper articles from as early as 1863. From "The Great Match: Heenan and King," in the New York Clipper (April 25, 1863):
This is but a fulfilment of the programme which we alluded to when Heenan last departed for Europe. It is quite likely that Henan and Sayers will draw immensely for the concern they travel with draw immensely for the concern [Howes' Circus] they travel with, and when the tenting season is over, it is probable that Sayers will assist Heenan in his training, and be in his corner on the day of the fight. After that important event, the two may make a tour of England, Ireland, and Scotland, and the Boy will then bring the gallant Sayers with him to this country, where a cordial welcome awaits him.
Early examples of figurative usage
Figurative usage is evidently at least a little older than Ammer supposed.
From "King Cotton Is the Redemption of Hope," in the Carlsbad [New Mexico] Current (December 22, 1922):
It goes without saying that we have been hit hard with the great drouth, the bank wreck, the slump in livestock. and the financial distress of the stockmen.
The Hope country has been tested and tried as never before and be it said to the praise and honor of the country and its people, we are still in the ring—are in our corner—and still have both mits on.
This is an explicitly boxing-related metaphorical usage and involves not someone being in someone else's corner, but someone being in their own corner. Still it marks a departure from instances literally involving boxers and their supporters positioned in corners of the boxing ring.
From "Withdrawal of Rit[c]hie Cheers Smith Forces," in the Greencastle [Indiana] Herald (June 19, 1928):
Withdrawal of Governor Albert T. Ritchie of Maryland as a contender for the Presidential nomination has cheered the [Al] Smith forces although they were confident long ago that the Maryland executive would be in their corner at the proper time.
This is a purely figurative instance and, like most of the other early instances, involves a political contest, which it equates with a prize fight.
From Muggins, "X-Ray and Sugar Candy," in the [Sydney, New South Wales] Sunday Times (July 1, 1928):
This story beats the band. Tom Walsh, whom the present Bruce Government was elected pledged to deport, has turned from strife, and preaches "Peace! Peace! Be calm," to the industrial insurgents. He has been successively derided as a Bolshevik and Sinn Feiner: yet, to-day, those who then condemned him as a dangerous firebrand, whom Australia must deport for Australia's good, are in his corner, reciting Bracken's lines, "Not Understood." Walsh stands four square and denounces his erstwhile confreres as enemies of Britain and British good; while those who yestre'en chanted "For He's a Jolly Good Fellow" now roar opprobrious epithets at him, and look upon him as a n industrial apostate and renegade. Verily, Peace hath her defeats no less disastrous than those of war.
From "On the Way" State Lottery Bill Likely" in the [Perth, Western Australia] Mirror (November 8, 1930):
The opponents of State lotteries are only a very SMALL IF NOISY MINORITY the same killjoy minority that objects to Sunday charity carnivals and mixed bathing and most other things except church. And the Upper House isn't likely to present such a lot of opposition for the temper of the people in general is obviously turning towards a State lottery.
Sir James should give it a fly. If he does "The Mirror" promises to be in his corner.
From "In This Corner..." in the [Urbana, Illinois] Daily Illini (January 20, 1933):
President Hoover and President-elect Roosevelt meet again today in one of their historic pre-inauguration pow-wows on world affairs. Both men will again have their "seconds," and, as the radio announcers say, they are both "in the pink of condition" for the battle. We do not see any need for parceling out the so-called "seconds" between the two men as if they were boxers going into the ring together, but the news services have it that Secretary [of the Treasury] Mills and Secretary [of State] Stimson are scheduled to hold the bucket and towel for President Hoover, while the President-elect has not named the man or men to be in his corner.
From Ray Tucker, "News Behind the News," in the Santa Cruz [California] Evening News (June 15, 1938):
Said one of his [Representative Homer Smith's] managers in discussing the Townsendites' trek to Thomas: "Why, we can buy those babies for two bits apiece. All they want is money. Once we give them a look at real cash, they'll be in our corner."
From Raymond Clapper, "New Deal Jolt: Numbed by McNutt Choice," in the Santa Cruz [California] Sentinel (July 15, 1939):
From all of this there is one notable exception—Secretary [of Commerce] Harry Hopkins, who is much taken with Governor McNutt and apparently is in his corner. I suspect Secretary [of the Interior] Ickes is the most critical. He probably feels it more because he has led the crusade for a third term [for President Roosevelt].
The expression "in [someone's] corner was in use in the context of boxing matches at least as early as 1863. It begins appearing in contexts where the usage is purely figurative no later than 1928, in both the United States and Australia. Not surprisingly, given the similarities between prize fights and political contests, early figurative use often seems to have involved elections or other political controversies.