What they are basically saying is that viruses can use the Bitcoin blockchain to communicate with their authors. So for example the virus author could put code into the blockchain and the infected computers would all get that code from the blockchain and run it.
This is a concern to kaspersky because normally the viruses would connect to a server, called a command and control server, to receive new instructions from the virus' author and send back stolen data etc. So all law enforcement would have to do is shut down the command and control server and they can cut the virus authors access to the infected computers. However, if the virus was using a blockchain, there would be no central point of failure and cutting the authors access would be non-trivial.
Other security researchers also had concerns about the website pastebin.com for similar reasons, that it could be used for botnet communication:
http://blog.spywareguide.com/2009/06/pastebin-botnets.htmlMost people are going to read this article and take it to mean that computers can be infected via the blockchain. This is not true. What they are talking about is using the blockchain as a way for hackers to send instructions to infected computers.
"Other security researchers also had concerns about the website pastebin.com" I've used pastebin to communicate between personal computers, but I don't think it would be a good example because in most case the program would be depending on one link for instructions, if it's removed then the program is vitally dead.
My main question, was why didn't they exploit the bug (using the blockchain) when they found it, until you explained it.
From my understanding the only solution, since you cannot restrict the type or format of data included in the blockchain, would be to update their antivirus to monitor the behaviours of local programs which listen for data included on the blockchain, then proceed to quarantining these programs.
So the problem is not that there's a code which can harm the Bitcoin ecosystem, but virus owners could utilise the blockchain to communicate with their Trojan horses...