Once again, you don't have to ask players to download Java because most likely they have it installed already. And to be sure, you can use a utility to bundle the Java runtime with your game. Just search for one online. Spurious warnings from antivirus software are common, many developers run into that problem. And Java applets were deprecated precisely because they were terribly insecure.
No, I don't use Eclipse, but I did make a Java applet long ago, and it has to be made as an applet, which inherits from a specific class in the standard library, and draws on a canvas in the way of output. Eclipse being a big serious Java IDE probably has a project template for that kind of thing. Look for it, and/or check the Eclipse documentation.