My Xbox 360 controller works fine, and this is the first thing I've read about anyone having problems with non-functioning controller. You don't even need to 'enable' the 360 controller in Costume Quest, it just works; most in-game instructional tutorials are even accurately depicted for the 360, too.
You probably already know this, but never hurts to re-cover the bases: Have you used the controller with other games and it worked? Did you plug in your controller before starting Costume Quest? Is the green 'Player 1' light on the controller lit? Is it a wired USB controller or are you using the wireless 360 controller? Generally, the USB controller is plugNplay, but the wireless controller needs drivers.
Here's my Input Config file: Input.lua. You can try replacing your input.lua file with this one (it's setup for Xbox 360 controller).
Go into: Program Files > Steam > steamapps > common > costume quest > Data > Config
Copy the existing input.lua file to another location, and replace it with the file you downloaded from the link. It may or may not help, you're welcome to try.
Sorry I can't be of more help, hopefully something as simple as re-starting your game will resolve it.
It seems to be working now: I unplugged my wireless receiver and searched for the latest driver. I noticed I already had the latest driver and plugged it back in. It seems like the unplugging and plugging worked. The other weird thing I have noticed is if I start playing the game then turn on my controller (by hitting the xbox home button) after the start, the controller doesn't work. I have to already have it on before the games starts. Windows does many strange things but I am all up and running now. Time to enjoy some Costume Quest thank you for the reply/help.