Launch Safari and go to Preferences > General
At the bottom of that window there is a piece about "safe files"
Take the check mark out of the box, close Preferences and Safari.

All files, by default, are downloaded to the Downloads folder.
Applications for the Mac usually arrive as ZIP files or as DMG files.
Double clicking either of them should open them with the required program.
By default, ZIP files are opened with Archive Utility and DMG files are opened with DiskImageMounter.

Before we move on, what type of files are you downloading?

If it ends with .exe then you are not going to be able to open them on your Mac, without assistance, as they are Windows programs.

Let us know