In-universe the cape belies his lowly (middle-class) upbringing. The Rogue One: Ultimate Visual Guide highlights that he's...
not of the principled Coruscanti classes, able to verbally parry and weave in
and yet his cape is described as an
Aristocratic tailored cape.
All of which suggests a man who's trying very hard through his clothing choices to portray himself as being of a higher class than he really is. Of course, Tarkin and Vader see through this immediately and treat him as a useful subordinate rather than as a potential equal.
This is backed up by the Star Wars Made Easy factbook.
WHY DOES HE WEAR THAT COOL CAPE?
Unlike most of the high-ranking officers in the Empire, Krennic isn't
a posh guy from a privileged background. He probably feels like the
cape gives him a touch of class. And given that Darth Vader wears one
all the time, no one's really going to argue with that, are they?
Star Wars Made Easy: A Beginner's Guide to a Galaxy Far, Far Away
Out-of-universe the design choice was picked to distinguish Krennic from Vader, yet pay homage to the original cinematography
StarWars.com: I asked Gareth about connecting Rogue One visually to A New Hope, and he mentioned how in the opening scene he used an
inversion — in A New Hope you have Darth Vader flanked by white
stormtroopers, and here you have Krennic in white flanked by black
death troopers. Was there more of that throughout the design process?
Doug Chiang: Yeah. I love those inverses, because Gareth always described Rogue One, the storyline, as the inverse of Episode IV.
Whereas in Episode IV, you had a farm boy who dreamt of being a
warrior, for Rogue One, it was the warrior who dreamt to go back to
the home life. So you already have, story-wise, a reversal, and even
character arc-wise, a reversal. So it was an idea to play with that
visually, and it’s a fun thing because it’s a motif that’s in there.
In my mind, when you layer those in there, it adds more depth to the
“WE SET THE BAR SO HIGH”: DOUG CHIANG ON DESIGNING ROGUE ONE
How did Gareth [Edwards] make sure this movie synced visually with Episode IV,
which was made 40 years ago? Making inversions was a big tool.
“[Episode IV] started with a guy in black surrounded by guys in white.
We do the opposite.”
ROGUE ONE PRESS CONFERENCE AT LUCASFILM – LIVEBLOG