That means the DNS server must be patched, or the company should upgrade to a more robust server infrastructure. One possibility is DNSSEC, which the U.S. federal government just deployed, but that is pretty complicated.
Another important thing is to make sure that all upstream ISPs have their act together. Even if the company's system is fine, dealing with any compromised name servers is disastrous.
In terms of where the responsibility of DNS security should reside, that depends on what kind of operational responsibilities the security team has. Many security teams these days are more influencers than implementers, which means they need to work with the organization's network team, which would actually deploy any remediation.
This was first published in July 2008