There is almost never a good reason to allow anyone to remotely access your computers. There is never a good reason to allow someone who has contacted you to remotely access your computer.
If you get a message about your system being infected, your best bet is to download and install Malwarebyte and scan your system with it. It's free! It's one of the top anti-virus and malware detection and removal programs out there. You can get the paid version to have 24/7 protection, but if you simply need to scan and clean your system, it works pretty well. Regardless, you really need to be sure you have a top rated AV (and preferably a full suite) protecting your system. This is more true today than ever with people and businesses being hit with ransomware.
That being said, no single AV program catches everything. Personally, I am using Bit Defender as my primary AV. I use Malwarebyte to do additional scans now and again. This gives me much better overall coverage.
The next thing you should consider is using a password program such as LastPass or Keypass. They do similar things, but in different ways. Keypass is good for keeping your passwords, usernames, and websites, as well as notes you need for pins and security questions. It can be encrypted and will even set notices for when passwords will expire. In my case, I keep my keypass data on dropbox, that way I can access it from any of my systems including my Smartphone. It uses a master key to get into it.
LastPass can be a desktop application, or an addon to your browser. Mine is set up as both and I have it set up to require authentication via my smartphone. It, too, can be used across multiple systems, but you have to pay for it to do that. The up side is that LastPass will create strong passwords for your sites, if you want it to.
In both cases, you can crate strong passwords, have portability across your systems/platforms and remove the not so secure saving of your passwords in your browser.
I highly recommend everyone using one or the other, and stop using your browser's built in password system.