The most common way I've seen is different .js language files being kept in a lang/ or locale/ folder in the project, with only the file for the current language being included using a <script> tag.
If you want to manage your languages in one, central, server-side place, I think you would indeed be best of outputting your translations from ASP as a JavaScript file. I don't know ASP but that should be perfectly possible to do - just remember to send a Content-type: text/javascript header along.
Related, PHP oriented but doesn't really matter: this question. Check out the translation plugins mentioned there, they may save you work additionally.