Sometimes simply restarting your PC will fix the problem.
Otherwise, go to Add/Remove programs and uninstall Java. (It's probably listed as J2SE Runtime Environment). Then download and reinstall from here: http://www.java.com/en/download/index.jsp
Bit of a longshot, but it was not working on my iPod Touch, so I cleared my Cookies and now it works again. They have recently re-designed the iPlayer area, so maybe the old cookie was no longer accepted.