Perception? The process of becoming aware. "Yes.. I am aware Utah has a good team this year". Not sure how that affects BYU.
Popularity? Fans are going to Fan. Fair-Weather fans may jump teams, but BYU losing has more to do with that than Utah.
Number of Fans? BYU's fanbase is not regional. If you are not in Utah you probably don't care (that's a PAC-12 problem).
Financially? BYU gets ZERO money if Utah wins or loses.
BYU and Utah are no longer linked. They have history, they might be rivals. But at most it's just one game a year then they go their separate ways. Utah would still be playing for the PAC-12 championship even if BYU beat them this year. BYU would still be going to the Hawaii bowl.