The BEAST attack touches encryption protocols that came before TLS v1.1.
Basically, the attack would permit an attacker to try and guess the part that precedes the real message, and thus guess the real message. Since the explanation is very technical, I'll let technical people speak on the issue:
edit: please do not apply what is written in this article. RC4 is not a viable solution. please do continue to support only strong cypher suite and force your users to upgrade to more recent browser versions if security is an issue.