I'm having the same issue. The behavior is not normal given that the "Browse" button does NOTHING at all when clicked even after the page fully loads.
What I found is that if I disable the Silverlight plug-in, the "Browse" button works. I have:
Firefox 3.6.4 (but it happens in 3.6.3 as well)
Silverlight Plug-in 4.0.50524.0