Open Firefox and click on Tools Options and then the Privacy tab.  Under Private Data check the box “Always clear my private data….” and uncheck “Ask me before clearing …” then click on the Setting button.

In the Clear Private Data box check “Download History” and any other download add-on you might have which appears.  For instance you can see I have Down Them All enabled, now click OK.

Finally while still in Options, click on the Main tab and under Downloads, uncheck “Show the Downloads window …” and click OK to get out of options and accept all the changes.

Viola!  Now you will not see the Download box pop up and every time you close out of Firefox all download history is cleared.

As I previously mentioned, I came up with this post as a means to rid what I consider an annoyance.  On the other hand, this is also a great privacy and security tip.