He made his debut in the Russian Premier League on 17 March 2013 for Zenit St. Petersburg in a game against FC Mordovia Saransk.

Baburin entered the pitch after 10 minutes of play when Yuri Zhevnov who started the game picked up the muscle strain.

[4] On 25 February 2021, he moved to Russian club FC Krasnodar, on a loan deal until the end of the season.

[6] On 21 June 2023, Baburin moved to FC Torpedo Moscow on a permanent basis and signed a three-year contract with the club.

This biographical article related to a Russian football goalkeeper is a stub.