Even if you've subscribed to the premium service there is no download function. That's even if you access your account from your PC.
Recently YouTube decided to change that by adding the download function to the desktop experience so that it mirrors the app-based experience.
The new feature is already in place. So if you are a premium subscriber all you've got to do is to click on the share options below the video. Another option is alongside the "three dot" menu and you'll see the new download option. Any videos you download from YouTube will be stored in your offline library where you can organize and sort them as you see fit.
Note that you'll also be able to specify the resolution of the downloaded video and choose between 144p to 1080p. Significantly there is no 4k download option available. The good news is that there doesn't appear to be any file size limitations beyond the available space you have on your hard drive.
The new feature is not browser specific. So whatever browser you use you should be able to download YouTube videos without difficulty. Again assuming that you're a premium subscriber.
It may not be a feature you use terribly often but it is handy and convenient and when you need it you'll probably really need it. Kudos to Google for continuing the good work of improving overall user experience and for making that experience seamless across multiple platforms.
It may not be something that will be sufficient to prompt most people to pay for the subscription service. However if you're already paying the monthly fee it's a nice perk that hasn't been available to desktop users until now.