This would probably only be done in source editor, since the alt text is hidden away in a popup menu in Visual Editor.

Even in source text however, I think it's likely that users will leave the alt text blank, which is counter-productive.

Not sure if there's an effective way to implement this with the current functionality. If we were to expand the copy functionality with a pop-up menu that lets a user fill in a form and generate boilerplate wikitext, then I think that would allow for supporting additional fields like alt text.