While I was reading a news article, a pop-up window appeared with a warning that someone may have assessed my personal information on the computer. It didn’t say clearly what I should do step by step. It just said something about downloading something and that I should not try to do it manually. The warning sounded rather dubious and I shut down the pop-up window and turned off the computer. Should I be worried?
That’s ransom ware. If you click the link it will download malware on your computer. You did well not to click on it.
You should run CCleaner once a month or so to get rid of cookies etc. I don’t get those kind of pop ups anymore. CCleaner is free and you can search for it on the net.
Wow, they finally got to “virus-proof” Macs? 
Jokes aside, it does sound like a ransomware popup. I am not a Mac expert, but on PCs, clicking to close that window could start the process of ransomware download. The Ctrl-Alt-Del option was a better way (although I gather that is not a 100% sure way to prevent its download). I would run a scan of the computer ASAP.
Must have. Should I report it to Apple?
@OspreyCV22 Is CCleaner safe? Guess I am a bit paranoid.
It’s not an Apple issue. You should be able to quit that tab because Safari spins each tab into its own process. Worst case is you force-quit Safari or use Activity Monitor to quit any particular open tab. In a few cases, the “bad” tab may try to reopen but you can then quit it or force-quit again and it will stop. This also happens with Firefox and Chrome. I’ve found it harder to get out of the pop-up mess with Firefox.
You should definitely read up on CCleaner so I am not your only source of info. I got it because it was discussed on another forum I’m on that deals with more tech issues. I don’t want to be in the business of hawking a product but it’s free and I’ve had it a while I also have Norton but it doesn’t catch everything. For some reason Norton always gives me a warning about this site.
^Ha! maybe I should wait until it happens again. Pop-up window kept popping up but I could close the tab and quit Safari.
Some of this ransomware first manifests as a popup or two, then the popups become more frequent, and then it becomes so frequent it renders the computer unusable. So watch out for this pattern.