Astronomy, and space science generally, is an effective method for broadening the human perspective, contributing new ideas and technologies and alerting the world to impending harm. After the unexpected Russian 2013 meteorite strike, astronomers at the University of Hawaii announced plans to build an early-warning system for such events, according to Space.com. Space science is crucial to communications satellites, GPS navigation and solar panel technology, states the International Astronomical Union.
Beyond the practical and defense-oriented uses for astronomy, the field is important to the modern world for the same reason it was important to the ancient world: astronomical discoveries have the potential to shift the human consciousness. Examples of this process abound in history. Close observation of comets and the phases of Venus were instrumental in disproving the geocentric model that put Earth in an exalted place in the universe. Learning the true distances between stars and between galaxies helped teach humanity more about its place in the cosmos. Searching for signals from extraterrestrial intelligence contributes to the human understanding of radio astronomy and even basic physics, suggests an article on the IAU website.
Another way astronomy contributes to the modern world is in developing the skills needed to advance other sciences. Modern astronomy can't function without advanced mathematics, precision engineering and sophisticated computer science.