Might help us help you if you told us your browser, borwser version and OS please.
Also please verify that you’re hosted here at wordpress.com and not elsewhere. Normally your username is linked to your blog here. If you are hosted elsewhere, you need to be over at http://wordpress.org as you would be running different software.
I’m having the same problem. When I go to My Dashboard -> Presentation -> Widgets,
I can’t drag and drop anything. I get the crosshair cursor, but I just endup selecting text. Using Firefox 22.214.171.124 on x86_64 Fedora Core 6. Funny thing is that on my FC6 i386 installation the drag and drop works. I suspect it has something to do with 64-bit Firefox.
Wordpress.com isn’t the only site I’m having a problem with, as I get the same problems with smugmug.com.