Well if you couldn't get into the game at all, then you should either try rebooting or perhaps reinstalling the game. Sounds to me like it could also be a shortcut error. If you have made a desktop shortcut or any shortcut for that matter that does not properly point to the doukutsu.exe file in the appropriate file folder with all the other necessary Cave Story files, then the game isn't going to start up.
If none of this solves your problem, I suggest you post your findings in a bit more detail, such as what does your computer tell you when you launch the game? What other programs have you been running while starting it, et cetera.