A few methods when it comes to applets.
A) As BubikolRamios pointed out signing is an option. Believe it or not it is actually possible to sign your own applets, but it generates a ton of warnings and things for the end user.
B) Run your applet and have it save data to a remote database. You can run database commands through a secure API you set up which validates that what is being transacted is valid. It is, however, possible to intercept and manipulate game data in this manner (but so is writing to a text file).
C) Make the applet an executable JAR/JNLP. Create a jar (Guide
) which can then be executed. When it comes to sensitive permissions the user will be prompted to give the OK to allow the applet to access system resources. Same goes for a JNLP, but a JNLP allows you to update the game content remotely and have it automatically downloaded to the person when they execute the file.