The observation in this answer that "batted down" in the context of a reporter's question may echo a U.S. football expression—"batting down a forward pass"—finds some support in newspaper search results. The earliest "batting down" in the sense of "knocking down" a ball that I found in an Elephind search, however, involves a baseball player knocking down (but not catching) a bounding ball (what today would be called a bouncer, a grounder, or an infield chop) with his glove. From "Griffs Divide with Boston, 2–1 and 1–0," in the Washington Herald (July 7, 1921):
It was in the second inning that the Red Sox made their lucky run. Ruel, the first man up lined to Shanks. Mogridge knocked down Scott's bounder, but could not recover the ball and the runner was safe.He caught him off first a moment later on a quick throw to Judge, but Owens missed the play. Then Foster's bounder was batted down by Mogridge, who deflected the ball toward Harris for an easy play, but Owens was in the way and the ball hit him, thus making both runners safe.
But six years later we see football players getting into the act. From "Illinois Is in the Lead," in the DeKalb [Illinois] Daily Chronicle (October 29, 1927):
Heston went in for Nymand and Dumhoff for Hoffman. Humbert fumbled on the Michigan 35 yard line, and Gabel recovered for Michigan. Grange batted down Rich's pass. Miller's line plunge and pass were smothered. Miller punted to Mills who was thrown on Illinois 21 yard line.
The batting down becomes figurative in "Valley Water Hopes Dashed: Assembly Battles Sales Tax Plan," in the Madra [California] Tribune (July 20, 1933):
The rural controlled senate ruthlessly batted down vital amendments, one, by Senator Jones, would have exempted foodstuffs.
And again, in "Allies Contest Red and Blues in UR Election" in the [Richmond, Virginia] Collegian (October 23, 1942):
The two offices that will be contested at the polls on Thursday are those of senator-at-large and secretary of the Senior Class. Some mix-up was committed last week when it was announced that Senior Class treasurer was the office in question, The theory was batted down by Willard Burton who stubbornly declared that he was the Senior Class treasurer and had no Intention of resigning.
The first instance I found of someone batting down a question from a reporter features a 20-year-old Lauren Bacall in the role of batter. From Jay Carmody, "'Keys of the Kingdom' Promises Bright Future for G. Peck," in the [Washington, D.C.] Evening Star (February 8, 1945):
Miss Bacall, draped in a breathless black dress, rises from her chair and slinks toward the door to retrieve an envelope that is edging its way into the room. (A Mr. Humphrey Bogart lives in a suite directly above Miss Bacall's.) The note is examined carefully and a slight frown crosses the actress's face. She bats down that unasked question with a slow blaze of words:
"And what's more, I intend to keep my private life private." The question shivers and hides behind a vase.
This selection of early literal and figurative instances of batting down doesn't do justice to the prevalence of football settings during the period (1921–1945) covered here; most occurrences of "batted down" involved that sport, even though the first such instance appeared only in 1927.
It may also bear noting that U.S. English speakers commonly use the kindred term "bat away" figuratively in similar settings involving someone warding off unwelcome questions. In U.S, football usage, "batting away" involves deflecting a forward pass away from its intended receiver in such a way that the ball goes upward or more or less horizontally for some distance before falling to the ground, whereas "batting down" implies swatting the ball more or less straight downward.
For example, my local newspaper today included an Associated Press story by Chad Day, "Trump Not Concerned About Manafort Cooperation," containing these paragraphs:
Trump told reporters at the White House that if Paul Manafort tells the truth to special counsel Robert Mueller's team then he doesn't see a problem. Trump also batted away a question whether he was considering a pardon for Manafort.
"I don't want to talk about it now," the president said.
The sense of the usage here is that Trump figuratively swatted away or deflected the unwelcome question.