[3][4][5] The New York Yankees selected Claiborne in the 17th round of the 2010 Major League Baseball draft.

[8] Claiborne began the 2013 season with the Scranton/Wilkes-Barre RailRiders, and was called up to the majors for the first time on May 3, 2013, when the Yankees placed Joba Chamberlain on the disabled list.

[11] He was then recalled to be the 26th player for a doubleheader against the Toronto Blue Jays on August 20 but was optioned back to Scranton/Wilkes-Barre right after the second game.

[12] Claiborne was optioned back to Triple-A on August 26 when the Yankees activated Derek Jeter from the disabled list.

He made 34 appearances out of the bullpen for the Double–A Richmond Flying Squirrels, posting a 2.38 ERA with 49 strikeouts and 3 saves.

[22] In 10 appearances split between the rookie–level Arizona League Indians, Double–A Akron RubberDucks, and Triple–A Columbus Clippers, he accumulated a 9.31 ERA with 10 strikeouts across 9+2⁄3 innings pitched.

[23] Claiborne was named pitching coach of the Hudson Valley Renegades High-A affiliate for the New York Yankees in 2023.

He was named pitching coach of the Tampa Tarpons, the Yankees' Single-A affiliate, for the 2024 season.