Windows is popular.
"The problems arise from closed source".
The problems arise from popularity, not from the closed source. It is a fact that windows products are much more popular then opensourced. Due to this there are lots of virus authors and other hackers efforts are devoted to finding bugs in Microsoft products, not in opensourced ones. I believe that many popular opensourced products have at least the same amount of bugs as Microsoft's products have, we just don't know about it. And even if a vendor or an opensource community will fix the bug rapidly it will not have any effects as it could be seen with the Microsoft's case: noone will care installing the bugfixes and patches! If you look at the history, you could notice, that Microsoft had released patches for some critical bugs weeks before the viruses exploiting those bugs appeared. In case of Linux community it could be even worse, because Linux does not allow that easy binary-patching as Windows software does, it is more difficult to install a bugfix for an opensourced program. And nothing will change if there is a bugfix, but it was not installed by the end user or a sysadmin.
"Alternatives to the Internet and email?"
"Safest of all would be to take up traditional communication methods like fax and postage again, instead of relying on email."
We receive quite a lot of fax spam and junk mails in our office, and our fax number is not even a well-known advertised fax number. Fax spam is even more expensive for the spam recipient than the email one.
"Secondly, the Internet could be partially replaced with parallel, closed company networks by using VPN (Virtual Private Network) technology. This would allow messages and information to be transferred securely inside a single company."
In our company we have some internal email system which is not advertised to the outside world. We do not receive any spam within internal email.
"A third alternative would be to restrict incoming email messages to those from known senders."
It is not possible to strictly identify the sender of an email without moving to SSL/X509 protocols. BTW, any web forms without anti-robot protection can be easily spamed and any anti-robot protection annoyes your customers.