In 2016, as a freshman at TCU, Baker was named the Big 12 Conference's Freshman of the Year.[10] He started all 67 of TCU's games, batting .379 with 11 home runs and 62 RBIs.[11] As a sophomore in 2017, Baker played 47 games before he suffered a left arm injury during a collision at first base that ended his season. In those 47 games, he batted .317 with eight home runs and 41 RBIs,[12] earning him a spot on the All-Big 12 Second Team.[13] Baker returned healthy in 2018 for his junior season, but his season was cut short again after 31 games after he fractured his left fibula and tore a ligament during a slide, requiring season-ending surgery.[14]
Baker spent the 2019 season with the Palm Beach Cardinals.[19] After hitting .346 with four home runs and a 1.067 OPS in the month of August, he earned Florida State League Player of the Month honors.[20] He finished the year slashing .244/.327/.390 with ten home runs and 53 RBIs over 122 games.[21] Baker did not play in a game in 2020 due to the cancellation of the minor league season because of the COVID-19 pandemic.[22]
Baker was assigned to the Springfield Cardinals to begin the 2021 season.[23] He missed nearly all of August and the beginning of September while on the injured list.[24] Over 91 games for Springfield, Baker slashed .248/.322/.530 with 26 home runs and 68 RBIs.[25] In mid-September, he was promoted to the Memphis Redbirds of the Triple-A East with whom he appeared in two games to end the season.[26]
Baker returned to Memphis to begin the 2022 season.[27] In 124 contests, Baker slashed .228/.289/.394 with 21 home runs and 66 RBIs.[28]
The Cardinals assigned Baker to Memphis to begin the 2023 season. In 54 games, he batted .313/.434/.641 with 18 home runs and 53 RBIs. On June 4, 2023, Baker was selected to the 40-man roster and promoted to the major leagues for the first time. He was slotted into the starting lineup as the designated hitter, batting seventh against the Pittsburgh Pirates.[29] In 33 games during his rookie campaign, Baker hit .209/.313/.314 with two home runs and 10 RBI. After the season, he won the International League Most Valuable Player Award.[30]
Baker was optioned to Triple–A Memphis to begin the 2024 season.[31]