What's the best way for determining whether the user's browser can view PDF files?
Ideally, it shouldn't matter on the browser or the operating system.
Is there a specific way of doing it in ASP.NET, or would the answer be just JavaScript?
Neither, none, don't try.
Plug-in detection is not the right answer. I do not have a PDF plugin installed in my browser (Firefox on Ubuntu), yet I am able to view PDF files using the operating system's document viewer (which is not Acrobat Reader).
Today, any operating system that can run a web browser can view PDF files out of the box.
If a specific system does not have a PDF viewer installed and the browser configured to use it, that likely means that either it's a hand-made install of Windows, a very trimmed down alternate operating system, or something really retro.
It is reasonable to assume that in any of those situation the user will know what a PDF file is and either deliberately choose not to be able to view them or know how to install the required software.
If I am deluding myself, I would love to have it explained to me in which way I am wrong.