If Utah does well, it helps BYU overcome the 'mid-major' lable - a little. I don't really buy that what one team does effects the others.
BYU can overcome it alone by simply winning but Utah doing poorly could be used as a basic argument against Boise, TCU, BYU, Nevada, Tulsa, etc., not doing as well when faced with a good BCS conf. schedule. (The Big East probably does not count at this point)
Now, each team does there thing, and every season is different. I think most teams hold their future and general success in their own hands. Boise is always ranked very high. They define mid-major. TCU as well. They seem to have overcome the lable to an extent because they simply win. Boise plays almost nobody week in and week out, but they consistenly win the big game.
BYU is not tied to Utah or any other team. Win and it solves for all of it.
Note: BYU will not go undefeated this season and if they only win the easier games on their schedule, and Utah goes 6-6? Yeah, that may reflect poorly on BYU, as well as back to Utah.
If BYU did somehow go undefeated? What Utah does, one way or the other, has zero effect on BYU. Just like when Utah went undefeated in 04' and again in 08', nobody talked about what BYU did, or didn't do. We are rivals but just not that tied together.