You're most likely looking at using a flash/silverlight/activeX control. The <input type="file" /> control doesn't handle that.
If you don't mind the user selecting a file as a means to getting its directory, you may be able to bind to that control's change event then strip the filename portion and save the path somewhere--but that's about as good as it gets.
Keep in mind that webpages are designed to interact with servers. Nothing about providing a local directory to a remote server is "typical" (a server can't access it so why ask for it?); however files are a means to selectively passing information.
It allows you to select directories. The multiple attribute is a good fallback for browsers that support multiple file selection but not directory selection.
When you select a directory the files are available through the dom object for the control (document.getElementById('ctrl')), just like they are with the multiple attribute. The browsers adds all files in the selected directory to that list recursively.
You can already add the directory attribute as well in case this gets standardized at some point (couldn't find any info regarding that)