Generally, your best bet is to drop the author a polite note asking if/why the story has been discontinued. If the answer is bad or no reviews/feedback, flattery may work: tell that author that you really enjoyed the story and that you would like to see it continued. Don't--don't ever--phrase anything in a way that might be remotely construed as a demand, because that almost always makes people angry. Gentle pleading may help, but don't be too pushy.
If the problem was lack of inspiration on the author's part, some other story jockeying for mindshare, or, even worse, Real Life pressures, then there's probably nothing at all you can do, although flattery may still encourage the author to take up the story again when the problem condition has passed off (if it ever does). Above all, be patient--the author doesn't control these things, either. Accept that some fanfics just never do get finished.