When I select "Upgrade my pledge to the $50 level" I get a new section called Payment Method, with "Add a payment method" in red above a button reading Add a Payment Method (which is all a little redundant, but that's not the problem). The button doesn't actually do anything.
I had the same issue, it seems to be an issue with internet explorer. Try changing your browser and see if that works.
Firefox 16.0.2. But I'll try another browser.
EtA: It worked in Maxthon 1.6.3, which is a shell for the IE engine (and I have 9.0.11 installed).
Update: It worked, the $20 charge was processed (shows up in my checking account history), and my pledge was upgraded -- but the payment doesn't show up in my Order History. Well actually, there's a line for what I got as a result -- Thornkeep Print Edition (order #2322289) -- but it shows $0.00 instead of $20.00. Not a big deal, just thought I'd mention it.