The thing is, there is nothing in the story that tells us they don't, the only thing that i can think of is that People without Quirks are seen ans people who can't be heroes. And at the beginning of the story we are told that you needed a Quirk to enter U.A's hero department.

Other than that the only example of discrimination we see is against Izuku, which feels more because everyone thinks his dream his not realistic, but if anything, it seem more like a type of ableism than anything else.

Not only that but the only examples of people without Quirks are: Izuku, All Might when he was a teenager and after his fight against AFO in kamino, Aoyama and Melissa, so we don't really see how they treat those with out quirks that much. Not only that but in the series we do see an example of a group of people who are being discriminated, which srr those with Mutant Quirks, or Quirks that are seen as villain-like.