I'm going to teach you the most simple, effective, and safe way to download this game and play it offline whenever you want.
What you need:
-Mozilla Firefox
-ScreenSwift
How to:
Step 1: Go to http://www.newgrounds.com/portal/view/498969
Step 2: On Firefox, go to Tools > Page Info > and Media
Step 3: Scroll down until you see the address "http:// uploads.ungrounded.net/498000/498969_Super_Mario_63__2009_.swf" (Type is "Embeded")
Step 4: Save it.
Step 5: Download and install ScreenSwift.
Step 6: Using ScreenSwift, under "General Options", make the "Flash Movie file path" the file you downloaded in Step 4. Name the ScreenSaver "Super Mario 63"
Step 7: Under "Exit Screen Saver on:" Un-check everything except "Show close button."
Step 8: Make the screen saver background color black. Leave everything else alone under "General Options".
Step 9: Under Movie Options, make the movie quality Best and the size "Keep original size". Leave everything else under "Movie Options" alone.
Step 10: Click "Installer" on the left and choose where to save the installer.
Step 11: Save the installer. Exit ScreenSwift. After saving the installer, open it.
Step 12: Click install.
Step 13: After you install it, you can now go to your Screensaver options under Destop Properties and "Super Mario 63" should be in the screen saver list.
Step 14: Click it and press "Preview". This should run the game in full screen and you can exit by clicking the X at the top left.
Hope this helps! If you have any trouble or questions, just ask. Enjoy!