SKSE download not working

Solution 1:

It's likely the file has actually been downloaded, but you have the '(Always) open after downloading' option turned on, and Windows tries to open the .7z file.
You can check your default Downloads folder (e.g. C:\users\\downloads) to see if this is the case.

As the SKSE website says right underneath the files:

Having trouble extracting the archive? Install via Steam, use the installer, or download 7-zip.

Just download the open source 7-Zip for Windows (likely 64-bit, otherwise 32-bit) from that website, and you should be good to go.

For classic Skyrim you can actually download an installer, that you will be able to open straight off the bat. You can find it at the top of the SKSE website.