As quality drops, so does circulation. Maybe they'll print it with the name of a famous author to boost sales. If a magazine put, say, John Grisham's name on your story, Grisham's attorneys would be on the line within a week. Besides, names don't make that much of a difference to a magazine. In addition, one of the joys of editing is discovering a new author.
Editors are delighted when they can publish someone for the first time.
OK, so they may not steal my story. It's the execution of that idea that makes a story. A brilliant idea is worthless if the story is poorly written, with weak characters and no plot. Similarly, some excellent stories have been written from very unimpressive ideas. Since the chances are quite good that someone thought up an idea similar to yours independently, you can't depend on ideas to succeed as a writer.