Should be something here for you:
https://www.chess.com/forum/search?keyword=how+to+download+a+game
You can't download them all in one shot, but you can certainly download. Maybe this discussion will help:
I think I am already using the latest version of the site. Please take a look on the screen shot which shows the home page as it looks like in my PC.
If this is Version-3, it is displaying 25 games in a page as I mentioned earlier. Please tell me if there is a way to increase the number.
Yes, that is v3. On the game archives there is a More link that can be used to load more games. All previous games stay on screen so you can do that a number of times then dowload all showing.
There may be a limit with that but I haven't heard that is the case.
Like YO! I like to study my games on my other chess programs (read- not on chess.com). To this end how do I download my games??