Baron attended John A. Ferguson Senior High School in Miami, Florida. He committed to play college baseball at Duke University, but was selected by the Seattle Mariners with the 33rd overall selection of the 2009 Major League Baseball draft.
In his first professional season, 2009, Baron played for the Rookie-Level Pulaski Mariners of the Appalachian League. He batted .179 with 19 hits, six doubles, two home runs and 13 RBIs in 30 games. In his first full season, he led all of minor league baseball in catcher caught stealing with 46%. He was considered one of the best defensive catchers in the '09 draft and was awarded the 2009 Rawlings Gold Glove award. Baron made his MLB debut in 2015. He is one of three catchers in Major League history to catch a one-hit major league shutout in his first ever start. He was designated for assignment by the Mariners on November 2, 2016. On November 10, he was released. Baron signed a new minor league contract with the Mariners later in November.
He elected free agency on November 6, 2017, and signed a minor league contract with the St. Louis Cardinals on December 22. He played with the team's AAA affiliate, the Memphis Redbirds in 2018, until being recalled on May 17 due to Carson Kelly being placed on the disabled list. His first major league hit, a single, came on May 19, 2018, off of Luis García of the Philadelphia Phillies.

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