There are many different things you can do, but it all comes down to: do your programmers care? If they don't at all, it will be impossible to make them improve. If they care a lot, they will improve themselves. If they care a little, that's where you can try some of the strategies others have given here.
As for my specific suggestions, I would recommend "The Pragmatic Programmer: From Journeyman to Master", and actually apply the principles in the book yourself. Use unit tests, setup CI (such as Jenkins), a code review system (such as Gerrit), a bug tracker, code analysis (such as Sonar), and use tools (linters, etc). Also, cranking up warnings on your compilers/interpreters and not allowing in anything that doesn't pass muster is a good start. This might all seem like a lot, but you can do little parts of it at a time. It is worrisome that you say you can't review every commit. Perhaps you could assign members of your team to review each other's commits? This would also have the bonus of giving them a feel for what it's like to be on the receiving end of bad code, and to teach each other.
As for my specific suggestions, I would recommend "The Pragmatic Programmer: From Journeyman to Master", and actually apply the principles in the book yourself. Use unit tests, setup CI (such as Jenkins), a code review system (such as Gerrit), a bug tracker, code analysis (such as Sonar), and use tools (linters, etc). Also, cranking up warnings on your compilers/interpreters and not allowing in anything that doesn't pass muster is a good start. This might all seem like a lot, but you can do little parts of it at a time. It is worrisome that you say you can't review every commit. Perhaps you could assign members of your team to review each other's commits? This would also have the bonus of giving them a feel for what it's like to be on the receiving end of bad code, and to teach each other.