Theory of the BEAST Attack
SSL uses various combinations of public and symmetric key algorithms. (OpenSSL's list) When the symmetric algorithm is a block algorithm (as opposed to stream), cipher block chaining is used -- the previous block is part of the input to the new block.
Why are we still suffering from this
We should be getting away from TLS 1.0 and moving on to 1.2. But of course, we still live in a world with SSL 2. More to the point, even where you can switch to newer TLS versions on your server, the ubiquitous NSS library doesn't support it. That means you wouldn't get Firefox or Chrome visitors.
You can use stream ciphers (RC4), and that will prevent the attack. I don't know if the empty opening message from my first link is used at all, or how that would affect various client implementations.
Your personal reaction
Help write TLS 1.2 into the NSS library. Make others know that you consider security a priority. If you have the time, do some research, testing which non-standard SSL configurations will work and what platforms they negatively affect. If you find out that RC4 works with everything, publish that so we know it has been tested.
If you don't have the time or capacity for that... wait it out. That happens too often, but at least a few folks are being squeaky about it.
Move rc4-based cipher suites to the top of the list; the attack seems to be related to CBC, and rc4-sha is not vulnerable if that is the case.
Update - we've released a white paper here detailing how to make the changes. Available here. To clarify, protocol version won't help you here due to practical considerations - servers cannot disable TLS1.0/SSL3 at the moment due to poor client support, and merely enabling 1.1 still leaves you open to a downgrade attack even if the client does support 1.1. Going to a non-CBC cipher is the only effective server-side mitigation I'm aware of.
You could get a better insight at SANS Diary Entry at http://isc.sans.edu/diary.html?storyid=11635 the only possible way (discussed at entry above) to counter it at Server is mentioned as disabling the ciphers using CBC
For a better understanding of the BEAST, you could have access to the SSL Paper for it and the Java Code at http://www.insecure.cl/Beast-SSL.rar