Lots of bands in the 70s and 80s employed a horn section. I love funk, so I love horns, and I've noticed a trend where the saxophone gets it's own solo parts in a song (Superfreak by Rick James, Turn the Page by Bob Seger). I've also seen quite a few live shows where the sax player receives a disproportionate amount of applause. Clarence Clemons and Kenny G are arguably more well known than a lot of guitar players. Is there any reason for the rise of the sax, and can we trace the history of it?
First off, the saxophone is actually a woodwind instrument, not brass. It was designed to be a less cumbersome version of the bass clarinet.
It does blend very well with brass instrumentation, and it became popular almost immediately with military and concert bands.
If we're talking about pop music and not jazz ensembles or symphony orchestras or the like, here are a few reasons right off the top of my head:
The transition from "Big Band" jazz to bebop and other jazz forms that emphasized virtuoso sax performances. After WW2 you start to see less big brassy dance band music, and the artistic zeitgeist shifted to people like Charlie Parker, John Coltrane, and Dexter Gordon. There were a lot of other virtuoso musicians not playing saxophones, of course, but saxophone was more prominent at this time than it had been previously.
Popular rock musicians becoming interested in more "roots" music like jazz and blues starting in the late 1960s and into the 70s. You have the Rolling Stones recording in Muscle Shoals, Eric Clapton's ongoing blues influences, etc.
You really can't answer this question without honing in on Bruce Springsteen and Clarence Clemmons. Springsteen was coming up in the era of rock's return to its roots I mentioned above, and early in his career he styled himself more of an artist than a pop music idol. The saxophone was already associated with art music, the avant-garde, and virtuosity, so it's kind of a no-brainer that as Springsteen evolved his sound, he would heavily feature the saxophone.
Without knowing a ton about the careers of Bob Seger or Rick James, my guess is that pop acts like that were going into their labels and saying "WE WANT AN ALBUM THAT SOUNDS LIKE THIS!" in re Springsteen's sound. It's almost like a trickle down effect, where one guy uses the saxophone to sound sophisticated and unique, and then everybody else aping the style to the point of total dilution. (No idea where Kenny G comes into all that, since IMO he's not really working in the pop idiom, or at least not until much later, and in a more "adult contemporary" as opposed to rock landscape).
(OK so I googled "Turn The Page" because honestly I really don't know a ton about Seger, and it turns out that song and its iconic sax riff actually precede Bruce Springsteen's entire career. So I guess I'd put him more in the category of my second two phases -- rockers looking to roots music and innovators building that into a sophisticated new sound -- and less in the category of Rick James schlocky sound-alike. I know linking Wikipedia is frowned upon, but the wikipedia article on "Turn The Page" has some good quotes about the sax melody that might help answer this question for you.)