Since I only use my desktop for work, and to update my ipod a couple of times a month, I shut it down when I’m not using it. That makes me wonder, am I reducing my chances of getting malware or having other bad things happen to it through the internet, or would I also have to unplug the phone line that goes into it? Just curious.
I don’t think there is anything that can enter if the power is completely off from a wall switch or one with surge protection and an on/off switch like are used everywhere now.
Unplugging the other wire, phone or router wire is maybe overdoing it but if you turn off the router, if you have one, it will be more secure from lightning strikes. Both can be done in 2-3 seconds so if the peace of mind helps, do it.
I do several things that others laugh at but since I don’t ever have to
whine about bad stuff happening, it is worth it if I can stay like that.
Do what makes you comfortable, it won’t hurt and might help…
YMMV
No, you are not significantly reducing your chance of getting malware.
A computer sitting idle is not much of a target these days. Most malware comes from email links and malicious websites. If you are not surfing the web, then there is no difference between idle and off. As for email, well, you are going to download all your (potentially dangerous) emails the next time you turn the computer on, so that also makes the computer being off no safer than it sitting idle.
However, if you are not using it, having it off or asleep saves energy.
Basically, all you need to do is just close your browser when not in use, right?
I used to shut off the PC every night, and then I remembered I had set up the auto updates for my anti-malware programs to happen in the dead of night. Now, I restart every night to wipe the browsing history, and I don’t worry about it.
What’s a “phone line”? 
I did not think anyone still used that old style email client? Should have one that does not download anything until you request it.
IMO), there should be nothing that auto downloads on a computer.
Almost everyone I know uses an email client.
Web-based email is for email dilettantes.
Restarting doesn’t wipe your browsing history.
Depending on how “Restarting” is defined, it might.
Set options on browser to dispose of browsing history, empty cache, etc. when exiting.
Exit browser manually in the course of shutting down computer. Alternately, turn off exit confirmation prompt in browser so that shutting down computer successfully exits browser without hanging on prompt.
Restart wipes browsing history. Q.E.D.
Thunderbird on all my Windows and Linux desktops or laptops. For both a gmail account and an ISP-provided email address.
Webmail is for situations where you don’t have a configured client handy.
OTOH, I don’t run the client continually in the background, so it’s not constantly downloading emails either. It runs only when I check mail. Since I’m not receiving hundreds of emails a day, the delay between starting the client and reading any new emails is imperceptible.