I've been having this same problem with saving pdf files File - Save As from the top Toolbar on my laptop.
I would constantly get messages - There was an error opening this document. The file is damaged and could not be repaired.
I also downloaded the lateset version of Adobe Reader XI
But at last I have managed to find a simple way to access pdf files from the internet.
Firstly, without downloading the pdf, I printed the pdf file direct from the Safari page by using the Print icon on the grey lozenge that appears centrally three quarters down the screen.
Then I discovered that if I save the pdf using the Save icon from this same grey lozenge on the Safari page I can open and print this pdf version from my desktop.
I've tested it with a couple of different pdf files and these methods work using the grey lozenge icons but the same pdfs files don't work using File - Save As on the Tool Bar.