I feel bad for the sysops who're scheduling 3am maintenance windows on production TLS services for the edge case this presents. God damn security researchers!
Under the right circumstances, BEAST can be exploited with a real-time practical number of connections. It was demoed on stage. Those circumstances don't include every browser configuration, but they're not very extraordinary.
Does that answer your question? :-)
However, TLS is not only used between browser and web server. We can use better ciphers in situations where both ends can be controlled.