Personally, I'd eventually offer them an in-universe choice... be a Company Man working for a group like SK-Prime or Firewatch, keep rolling the bones knowing that they're high-profile and it's almost impossible to hide, or get a Reprint mod and some plastic surgery, then retire on their savings.
I also use it as a scale of how likely someone that's not a PIC (Plot Involved Character) has heard of them during any interactions. Roll 2d6 and it the result is equal to or below their Public Awareness, then they're known to the person. Straights* and people in areas you haven't operated in before roll +1d6, while those directly involved in the Shadow community roll -1d6. And yes, if you operate out of Seattle and you've just taken your first jaunt to LA, a local Straight would roll 4d6.
That means that, as you climb the totem pole and get more attention and you're more likely just get randomly spotted and tagged as you're walking around town. Disguises can limit the attention to the "working" identity as long as the player can keep the connection secret, and of course burning a disguise and taking a new identity qualifies for the whole Street Cred / Notoriety trade.
And yes, that also means, barring other modifiers, you don't really even start making a name for yourself until the 30 Karma mark and it's 60 karma before you get added to any "watch lists" unless you're stupid/unlucky enough to get caught.
*Aka, wageslaves... people with no knowledge of the seedy side of the Sixth World.