With apologies, I do not have any answers for you yet, but I do have a few questions to help me and other folks better understand the situation you're encountering.
I'm going to guess you're using Finder to go to the downloads folder or your documents folder and then double-clicking on the files. What diagnostics or errors or messages do you see when you're prevented from opening the downloaded files?
What are the files? Are the saved files involved text documents, or are they application programs? (If you're not sure, what were you told the files contained?)
Are you running any add-on tools such as virus scanners or malware detection tools?