Go to previews and premieres. Many times, a film is launched or "premiered" in a number of different media markets, and the film's director attends some of the premieres. Try to get early tickets to premieres to watch the director's film with that director in attendance and get a chance to meet the director one-on-one after the film.
Find film hangouts. Francis Ford Coppola, for example, wrote "The Godfather" at Caffe Trieste in San Francisco, and Woody Allen frequently has dinner at Elaine's restaurant in New York. Find other director hangouts such as these to sit, eat and chat with famous film directors.
Join filmgoers clubs. Theaters, especially art-house chains like Landmark, offer subscriptions to film clubs where for a yearly fee you get to attend a certain number of films with directors and other members of the production team doing question and answer sessions after the screening. Take advantage of the opportunity to stand up and ask a pertinent question or make an insightful remark that can be the basis of further conversation with the director.
Enroll in film school. Prominent film schools usually have famous directors on faculty, even if the directors aren't teaching classes every day. While film school is a major commitment, it will bring you closer to film directors and their work. If you don't want to enroll in a full-blown degree program, look for lecture series or associates degrees at the schools.
Become a film critic or writer for a periodical or webzine. You will be able to use this credential to book interviews with filmmakers when their latest film comes out.
Work in the industry. Film is a huge industry that encompasses many different positions and skills. If you are passionate about meeting directors, get a job in the industry that brings you into contact with film directors on a regular or semi-regular basis.