Obviously the answer would be "a lot" since you are reliant on them publishing you, but how much control is typically exercised? I always thought that editors would step in and say "woah, no, you can't do that, it makes no sense" or something along those lines when an author starts writing something that makes no sense or reads like a 12 year old's fantasy in school (the bad kind). Does that actually happen, or do they mostly not care? Sometimes, I start reading a story, and the writing in it is so bad that it makes Twilight look like a masterpiece, no joke. Hell, sometimes the synopsis is so bad that it just makes me cringe and stop right there. And I can't help but think : What was the publisher doing? What about the editor? To get an idea of what I mean by bad, read the middle panel of this image :
I have no personal experience publishers and editors, so I can't really give you a real answer. But my immediate thought is that if a publisher has accepted a manuscript, they already think the content of the story is good enough - but may require a lot of tweaking to make it more readable.
Publishers don't Really care if the writing is good. They care if it's sellable. Crap writing sells, as long as it's the right crap writing.
In a sense, no control. You can always turn down the contract. If you want the publishing contract bad enough then they have some leverage. I know someone who rewrote portions of a novel and deleted scenes so the publisher could market it under a different genre than the author intended while writing it.
