OK, what I think is happening is that you used to have the checkbox 'Open "safe" files after downloading' ticked on Safari Prefs General tab*. When you open a .ttf font, the Fontbook utility (in your Mac's Applications folder) will open to load that font, which will then allow you to install it. But since that preference box apparently isn't checked any longer, Fontbook won't open automagically either. You'll have to do that manually by double-clicking the downloaded font file.
Another issue may be where Safari puts downloaded files like your fonts. You can see (and change) that on Safari Prefs General tab too.
*) NOT checking this box is very good practice from a security point of view.
PS, the request for a link simply would allow us to try things for ourselves with a file that caused you trouble.
