What are feasible quality criteria for Git repositories?
Who is responsible for checking that people don't check in commits that have large files? The same people who are responsible for checking that the commits aren't bad in other ways: everyone.

If git is new in your company, make sure that the git education includes this sort of thing. After that, it's up to code reviewers to notice mistakes and correct them.

If it becomes a repeated problem, then whoever is in charge of developer tooling can write some checks that run every night or whatever and let people know that something slipped through. But that seems like overkill in most situations.