How to get to know which app causes the "would you like to install java?" dialog in Mavericks to appear?
Since your screenshot references java where the app name usually is written, it suggests that it's the command-line version of Java that a given app is trying to run.

If it appears at specific intervals, it either an app that you have open or a periodical job running automatically in the background (Which, ironically, could be the Java Updater itself).

A quick way to check out which of your installed applications use Java is to run this command in Terminal (/Applications/Utilities/Terminal.app):

find /Applications -type f -name *.jar


This isn't 100 % accurate but it can give you a rough idea. Dreamweaver and Flash from Adobe, for instance, uses Java.

To get a list of currently running automated background scripts use this command in Terminal — You should run in as close as possible to the moment when the Java popup appears:

launchctl list


(Credit to this thread on Apple Support, it's also useful in understanding launchd)

This will, sadly, only give you a list of processes running and not why or what they're running.

Finally, you might want to take a look at the files in ~/Library/LaunchAgents (In Finder use Cmd+Shift+G and paste in the path. This directory shows the periodical jobs that you've setup as a user (sometimes indirectly through an install). Maybe one of these use Java.