Verified answer

A screensaver will not mess up downloads, and neither will turning off the screen. However, sleeping, hibernating, and shutting down will pause or cancel all downloads. Just change the power settings on your computer so that the screen shuts off, but the hard drive does not.

Screen savers don't affect your downloads. Just don't let your computer go to sleep or hibernate after a long period of mouse or keyboard inactivity so that the downloading won't be interrupted. If you like, you can use "Free Download Manager" to handle your downloading so you can also set the computer to shutdown after downloading is complete.

http://www.freedownloadmanager.org/

This way, you can leave the computer on if it's downloading a big file. It'll shutdown itself when finished. You can even stop the downloading if you have to turn off the computer for some reason. It will resume when the computer is restarted.