For that budget, you can choose just about any ATX mid tower. Most reputable cases at the upper levels have decent or better airflow. What's slightly more important is the case fans and what you do with them. What fans, how many fans, in what orientation, at what speeds, cooling what components.
For example, my fractal design define R5 has a silencing door on the front, so air is drawn from vents at the side of the door. I use the 2 stock 140mm fans for intake. Exhaust is the 2 stock fans on my nzxt Kraken X61. And that's it. Yet no matter what I do with my gtx970 and i7 3770k (4.6GHz) case temps never reach beyond 34°C, usually staying around 32°C. And the pc sits inside a cubby in my desk. I've got good airflow in a case that's...