To know with absolute certainty that they have simply installed a bookmarklet is not possible under normal circumstances.
To know with near absolutely certainty that they have installed AND USED a bookmarklet is possible. For example if your bookmarklet calls or navigates to a URL, you can add a parameter like &origin=bookmarklet
to the URL and track that on the server.
You could also get a pretty fair idea that a person at least tried to install the bookmarklet by using the drag
event on the bookmarklet link. Something like
<a ondrag="recordBookmarkletDragEventOnServerWithAjax()"
href="javascript:/* bookamrklet code */">myBookmarklet</a>
You could embed a unique ID number in each bookmarklet enhance your tracking installation tracking. For example:
<a ondrag="recordBookmarkletDragEventOnServerWithAjax('9h297gh3sdsd')"
href="javascript:id='9h297gh3sdsd'; /* bookmarklet code sends id to server */"
>myBookmarklet</a>
Using an ID number like above, it might be interesting to see how many people drag your bookmarklet but never use it.
Alternatively you could have them install an addon or extension which would could automatically make a call to your sever confirming installation.