Being online at any time is the problem, not being online most of the time - though that does, of course, increase the risk. Try putting a brand new computer online for the first time without the benefit of the firewall in your router. It takes no more than 30 seconds to start seeing automated hacking attempts to commonly used ports.
The only way to be safe is to have a never connected system. This is known as an "air gap". However, even this isn't that safe as the US Airforce found out when some of their UAV systems got infected after someone had opened an infected file. They were not ever connected to the Internet. Of course, in that case no damage occurred other than embarrassment and the time/energy required to cleanse all the PC's. But a targeted attack might have caused serious damage.
In your example, your supposed safe computer is actually very vulnerable (though not as vulnerable as the connected PC of course) due to you exchanging information with it from the online PC. To reduce this vulnerability, you would need to ensure that any data you passed from one to the other was carefully cleansed of any possible infection. The use of a Live CD would be sensible and would limit any infection of the connected PC. But you would not want to use a network file system to connect unless you had good protection of the network & even using USB would require very good anti-malware on the unconnected PC.
Then you have added a massive overhead to your computing environment so you would need to ask yourself if it was worth it. You would need to be working on very sensitive information, working on high-risk web sites, or be at high risk of attack to warrant that kind of overhead I would think.
