I’d also be wary of anyone who claims to have a definitive answer as to exactly how much, but I think it can definitely help, depending on the school. Here’s what the Columbia web site has to say on the topic, which is an FAQ:
http://mpp.music.columbia.edu/frequently-asked-questions
For what it’s worth, I know that the head of the jazz program at Columbia does talk with the admissions department about talented jazz musicians he’d like to see admitted. I also know that not all of them are admitted - I don’t think he has the same kind of sway that the head football coach has, but it is a factor they consider.
I’d also mention that, if you’re serious about pursuing jazz in college, Yale does not have a jazz program. Columbia has a fantastic jazz program (and, of course, it’s in NYC). Princeton also has a strong jazz program.
Best of luck!