Would it be good to double major in computer science and graphic design if I want to be a game designer?

Like Chris said, it'll help. And i'm sure there's a required class that'll help incorporate the story telling as you become a game designer. If not, it may be a good idea to look into storyboard art. There should be classes that deals with scripts for story telling in the field of a storyboard artist. The class should basically be an art of emotions and character in a story and i'm pretty sure it also teaches how to build emotion and character. Don't know specifically the name of the class but sharing the detail I just listed to a counselor should help... I myself is trying to double major, too! Architecture and Engineering. I feel I can get intrinsic satisfaction out of the two. I wish you a good journey!