The Dana Foundation in Manhattan has spent the past few years studying just this question and they've released their findings on their web site. The fundamental issue they address is "Are smart people drawn to the arts or does arts training make people smarter?"
The document is pretty long, but is broken up into interesting chapters such as "Musical Skills and Cognition"; "How Arts Training Influences Cognition" and the like.
From their web site:
Learning, Arts, and the Brain
The Dana Consortium Report
Learning, Arts, and the Brain, a study three years in the making, is the result of research by cognitive neuroscientists from seven leading universities across the United States. In the Dana Consortium study, released in March 2008, researchers grappled with a fundamental question: Are smart people drawn to the arts or does arts training make people smarter?
For the first time, coordinated, multi-university scientific research brings us closer to answering that question. Learning, Arts, and the Brain advances our understanding of the effects of music, dance, and drama education on other types of learning. Children motivated in the arts develop attention skills and strategies for memory retrieval that also apply to other subject areas.
The research was led by Dr. Michael S. Gazzaniga of the University of California at Santa Barbara. “A life-affirming dimension is opening up in neuroscience,” said Dr. Gazzaniga, “to discover how the performance and appreciation of the arts enlarge cognitive capacities will be a long step forward in learning how better to learn and more enjoyably and productively to live. The consortium’s new findings and conceptual advances have clarified what now needs to be done.”