I don’t think you really need to worry about anything.
Even a Java based game (of which they are few… Minecraft is the biggest one I know of) there still essentially no danger for you.
What the exploit means, is that an attacker can execute command on a system, by passing text through the logger… This isn’t something which is really going to happen when you are running software on your machine. It’s perhaps POSSIBLE, but not likely.
The real issue, is that Java still forms the backbone of most large scale enterprise systems, and a lot of those systems tend to have logging, for various diagnostic purposes… And most logging is done with this library, because it works well.
What makes this attack so effective, is that it’s really common to log user input, so that means it’s generally really easy for an attacker to exploit this, because they can just enter data into the places that the system is designed to accept user inputs.
To take Minecraft as an example, you can use this to attack the server, simply by entering the commands into the chat system built into the game… Because the server can log those messages.
But even as another use in that server, I suspect that the attack wouldn’t affect YOU, because your client is likely not actually logging that stuff, so it’s not leveraging that library for that stuff. (I may be wrong here, I’ve not dug into the details of it’s application in Minecraft, this is just my speculation) so while the server is vulnerable, you as a simple player probably aren’t. Now, if you run your own Minecraft server? THEN you may have a problem.
But for all the stuff that Java is more commonly used? Back end server infrastructure? I doubt that’s going to have any kind of real impact for you as a simple end user.
What’s more likely, is that an attacker will exploit the vulnerability, and then use that to do other shady stuff like steal your information if the guys running those server systems weren’t doing their other stuff right.
But I don’t think you need to worry about your own system somehow being attacked, simply by virtue of you connecting to some Java based system.