Very little. Look at it this way:
If you’re a prominent well-known person, your book sales are far more driven by your fame than what the publisher does. Sure, they will do significant leg work supporting the book, but nothing more than your executive assistant or publicity person does.
If you’re not a prominent well-known person, the publisher is very unlikely to put a lot of juice behind your book. They will get it out into the world, give it a little kick in the ass to start, but then wait to see what you do with it. This is why the marketing and promotions section of your book proposal is generally of the most interest to a publisher. The reason for this is very pragmatic — the publisher has many titles to manage at any given time; the author has just one. (By this I mean current titles in the active promotion. An author may have many titles, but really should be focusing on their newest.)
So, ultimately, a writer’s success is vastly due to her/his own efforts. This is something so many new and aspiring authors are shocked and disappointed to hear. They’ve naively come to believe that publishers handle promotions. Nope… “promotional support” is what I’d call it.
Think of it like this: If you started a business and got an investor, would that investor actively promote your business for you? Not likely, or at least not significantly. That’s not the investor’s job. He/she has put their dollars into your business; that’s their effort and risk. And to spread out and mitigate that risk, investors will typically invest in many businesses/projects, knowing maybe one out of ten will succeed and cover the losses or break-evens on the other nine.
This is exactly what a publisher is to an author; the publisher is your investor. But you are running the business of your book.