I don't remember a specific date being mentioned in the story, and I couldn't find a specific date either. However, the dates that the story likely took place during are quite limited. It's highly likely that the story takes place between 1929 and 1931.
The reason for those dates is that the story is based on Antoine de St. Exupery's experiences as an airmail pilot and director of the airmail airline, Aeroposta Argentina. Aeroposta Argentina began during the early 1920s, but expanded into South America (Argentina) in 1929. Placing the story before 1929 wouldn't have made historical sense to S. Exupery. The date of publication was 1931, and again it doesn't make sense for the author to put a book based loosely on his own experiences in the future.
Additionally the aircraft that Aeroposta Argentina used for mail service in 1929 was the Latécoère 25. That aircraft saw its first flight in 1926. Again for historical accuracy, there is no way that St. Exupery placed his story before 1926.