Step #1. Please check to see if you have a "Java" item in your System Preferences. The easiest way to see your System Prefs is to click on the black apple icon on the extreme left-hand end of the Menu Bar.
If you have installed Java, there will be an item/icon with that name in System Prefs. Click on it to open it. It will actually open it's controls in a separate window rather than listing various settings/controls the usual way.
Step #2. The first thing to check is a statement near the bottom of the "General" section. Mine says, "Java in the browser is disabled." What does yours say?
Step #3. Once you get the "Java Control Panel" open, click the "Security" label/tab. The first item is a checkbox for "Enable Java content in the browser". That box probably is not checked (or "ticked" as some Englishmen are wont to say!

). If that box is already checked, please report back:
A. Which "Security level" item is selected
B. Are there any entries in the "Exception Site List"?
I suspect Step #3 will solve the repeated requests for installing Java, however, please let us know.

.
Step 1. Java disabled for current user only
Step2. Java in the browser is disabled.
Step 3. Not checked