Apple has released an update for Java 6 on Mac OS X. This update brings with it important security updates, reliability improvements and compatibility improvements. This Java 6 update is available from Apple, while the newer Java 7 is available directly from the developers at Oracle.
Apple details the improvements to Java 6:
Java for OS X 2013-004 delivers improved security, reliability, and compatibility by updating Java SE 6 to 1.6.0_51.
On systems that have not already installed Java for OS X 2012-006, this update disables the Java SE 6 applet plug-in. To use applets on a web page, click on the region labeled "Missing plug-in" to download the latest version of the Java applet plug-in from Oracle.
Please quit any web browsers and Java applications before installing this update.
See http://support.apple.com/kb/HT5717 for more details about this update.
See http://support.apple.com/kb/HT1222 for information about the security content of this update.
Earlier this year, Java was compromised several different times through various different exploits. Many have decided that the world doesn't need Java on the web anymore with HTML5 and other solutions, hence many recommend disabling Java completely.
Depending on your operating system, you'll need to select the correct update or update your system through the Mac App Store. The update for Snow Leopard is available here and the Lion/Mountain Lion update is available here.