Attend a music school with a world music program. Several music schools and universities, including Wesleyan University, have world music programs with an Indian component, usually headed by at least one classically trained Indian musician.
Find a teacher by attending performances and expressing your interest to a musician, by contacting your local university's music department or by looking on boards and forums dedicated to Karnatic music. Take lessons, practice and educate yourself by reading books and watching videos the teacher recommends.
Go to India, attend concerts and find a teacher. The most outstanding performances, and the greatest concentration of Karnatic musicians, are found in the Indian city of Chennai.
Take Karnatic classes over Skype at sites such as Vidyasubramanian.com. Sites usually offer instruction for every level, sending class notes by email.
Attend a music school in India such as the Karnataka College of Percussion in Bangalore or the World School of Music in New Delhi (see World Music Central link in Resources).
Enroll in a U.S. based Indian music school such as New York's RagaChitra Foundation, with courses in Karnatic Indian classical vocal and instrumental music.