Not far at all. As the game introduces you into the world and characters at the start, ana (companion character) literally says something along the lines of, "if you wanna spend all the money you make from the showdown, you gotta make a name for yourself at night"(going off of memory).
Now when you say "mostly" then you should be fine. You can earn rep in the day by completing challenges which you can select in your garage, however you dont get much rep compared to night time races. However (spoilers.. not really) the main storyline missions happen mostly at night. While you can avoid the cops all you like, sometimes during races the cops will start pursuing you (its inevitable).
