First Aid checks the disk itself, it doesn't fix who owns your files, so whatever it cleared up was probably just the restart. Worth checking who the files belong to: click one of the stuck files, press Command-I, and look at Sharing & Permissions at the bottom. If your name isn't the one with "(Me)" next to it, or the owner shows as "system" or another account, something saved those files as a different user, and that's why Finder keeps asking for a password.
To fix the whole folder at once, click Downloads in the sidebar, press Command-I, click the lock at the bottom and enter your password. Make sure your name has Read & Write, then click the little circle with three dots and choose "Apply to enclosed items."
If it still comes back after that, whatever is saving into Downloads is the thing to look at. Does it happen with every download, or only files from one particular app?
— Dmitry · developer of ComfortFiles