Can I protect my rights to the story?
Your rights to the story? What makes you think you have rights to the story? And, obviously, you cannot protect something you don't have.
I'm afraid if I start trying to sell it, someone will take the idea and make it into a movie without giving me credit or compensation.
That's a concern that lots of folks have when trying to pitch an idea for a film. There are several ways to deal with this, but let's start at the beginning.
You told us that this story is contained in "a letter that was written to my great grandfather by his niece in Germany during WWI." I assume that means it was written by a person in Germany and that it was written sometime between 1914 and 1919. It also appears that the author of the letter was not a direct descendant of yours. Your great grandfather's niece was, presumably, a sibling of one of your grandparents and, thus, your great aunt. Is all that correct?
Setting aside the fact that the letter was apparently written in Germany by a German citizen,
section 303 of the U.S. Copyright Act provides that a copyright in a work created prior to January 1, 1978 but never published or registered for copyright for the term provided in section 302, but in no case earlier than December 31, 2002. If the work is published on or before 12/31/02, then copyright will last at least until December 31, 2047.
Section 302 provides that, in general, the term of copyright is the life of the author plus 70 years. That means that the letter is
possibly still protected by copyright
if your great aunt died less than 70 years ago (i.e., after 1947).
Even if the letter is still protected by copyright, the chances that you own the copyright are slim. I'm going to guess that your great aunt did not have a will that expressly provided for the transfer of the copyright in this particular letter. Therefore, the question becomes whether you could make an argument that you somehow inherited -- either directly from your great aunt or by way of one of your grandparents or parents -- the copyright. As noted, I think the chances are fairly slim -- especially if your great aunt was married and/or had children. Note that the fact that you possess the letter does not mean that you have any legal interest in the copyright.
On top of all that, the letter apparent tells a story that your great aunt did not create herself. Facts are not subject to copyright, but a factual story could be subject to copyright. But you don't seem to want to make money off the story itself. Rather, you want to pitch the facts as the basis for a movie (or whatever). That takes you out of the realm of copyright and into the realm of
idea protection, which is really nothing more than contract law. In a nutshell, if you pitch your idea to someone, you need some sort of contract (in some cases an implied contract may suffice) to protect your rights. If you seriously want to pursue this, I suggest you retain the services of a skilled entertainment/intellectual property attorney.