I'm making my game with a qwerty keyboard and the controls are designed around the WASD scheme.

How do games handle different kinds keyboards? Should my game detect the kind of keyboard the user has and change the input layout accordingly?

For the WASD scheme and similar, you should not check the keys using the letters on them to compare against, but the underlying key on that position (likely through checking scancodes). Thats much better than detecting which letter on which of hundreds of keyboard layouts the key got you need.

PS: Make it configurable, so people have a way out if your program doesn't work for them or their keyboard.