Technically, I think 'writing' down is correct, but you would hear more people say 'write' down. Perhaps someone more expert than I am can clarify whether I am correct or not.

"Mr. Smith plays games with us in class instead of having us write down everything he says." -- Correct. "Mr. Smith plays games with us in class instead of having us writing down everything he says." -- Incorrect. ""Mr. Smith has us play games with him in class, instead of writing down everything he says." -- Another correct way to say it.