While the source of malware is ultimately the Internet, it is possible to get infected from other computers on the same network. (Trust me, it's amazing how fast a worm propagates through a network.) Offline sources of infection include such things as autorun malware from USB sticks.
The fact is, a computer attached to a network cannot be considered secure. There are too many unknown zero-day vulnerabilities that can be exploited. However, you can reduce your risks. If you have a firewall you can block incoming connections from the other computers on your network. What you really want is defense-in-depth. A good software firewall that monitors incoming and outgoing connections, a good AV, locked down Windows, running as a limited user and using UAC to elevate to admin rights, encrypting important data, using strong passwords, minimizing threats by limiting software used, etc. Limiting the software you run reduces the "surface area" available to attack. Try to avoid Adobe software, as it is buggy and full of security holes. Keep the computer up-to-date.
Since you seem concerned with the security of your family's computers, why not offer to keep them clean, etc?
If you're really paranoid, create a known clean image and reimage the computer at each boot. You can also switch to a Unix-like OS, which are more secure than Windows in many ways.
When doing any online transactions, make sure that your connection is encrypted, preferably with AES-128 or AES-256. (This will be out of your control and will depend on the capabilities of your bank's servers.)