What's the downside to Quinn?

Honestly Quinns E is quite an overloaded kiting tool, with one ability you get a gap closer that puts you in exactly the right range, an escape that can go through walls, a knock back, a slow, and it procs the speed boost from W. Probably the best way to deal with Quinn is to play something that can interrupt her E, Draven can use stand aside for example, poppy would be the best at it, anyone with a line skill shot hard CC should be able to land it on her pretty easily. Really Riot kind of reduced the options they had for balancing her when they reworked her, having only 2 combat abilities and especially ones that reliable make her power curve overly reliant on numbers alone, like Taric or Skarner, too much reliability means they're either super powerful or never played depending on the patch.

Depends on who you are playing.

The last time I saw a Quinn she tried to abuse E in the mid lane against me. Unfortunately for her I was playing Viktor, and she kept popping like a balloon.

To put it simply E helps her escape and maintain distance against melee opponents, but it actually puts her in a more dangerous position against high burst, ranged enemies.

CC her in the middle of the animation. I've actually managed to trap an enemy Quinn under my turret as Soraka by putting my E under the spot she dashed from. When she bounced back, she got rooted and almost died to tower shots.