Borkowski was born in Wheeling, West Virginia to American parents, while all of his grandparents came from Poland. He and his family later moved to Steubenville, Ohio, where he attended Steubenville Catholic Central High School. He was one of the first in his family to finish high school.
He started playing the accordion at age nine, often performing at events in his community. In high school, he took classes in music and music theory. He attended Oberlin College from 1953 to 1957, earning a Bachelor of Science in Music Education. First, he planned on teaching music at the high-school level; however, mid-way through college he discovered his love of playing while studying with Reginald Kell. Then he began dedicating himself to the clarinet, with the hope of playing professionally. After Oberlin College he attended Indiana University (IU) to complete a M.A. degree. In December 1957, he began playing for the Indianapolis Symphony Orchestra while still attending IU. In 1959 he resigned from the Orchestra and joined the Jacksonville Symphony Orchestra along with his wife. After spending two years in Jacksonville, in 1961, he decided to attend West Virginia University to pursue a Ph.D in music education with a minor in musicology. He became a member of the national music fraternity, Phi Mu Alpha Sinfonia in 1963. While pursuing his doctorate he taught the university's marching band, working with the concert band, and taught clarinet. He completed the doctorate in 1967.
In 1967, Borkowski became a faculty member at Ohio University in Athens, Ohio. He taught music classes and conducted research. The next year he was elected to the faculty senate and later was chosen to serve as the assistant director of the School of Music. In 1970 he joined the academic area staff, the position of Assistant Dean of Faculties. A few years later he was appointed the Associate Dean of Faculties.
After Borkowski's tenure with Ohio, he was appointed the first vice chancellor at Purdue University in West Lafayette, Indiana. He was also Dean of Faculties for some time.
In 1978, he was appointed Provost at South Carolina University in Columbia, South Carolina. He was also Executive Vice-President for some time.
Borkowski was nominated for president at University of South Florida (USF) by a librarian. Several things compelled him to apply for president: the appearance of the campus, many opportunities, the status of being a young university with little tradition, and the university's fine arts program. After he applied, he was interviewed by a chancellor and got the job. He became the seventh president of the University of South Florida in 1988. During his tenure he had many goals: make USF a top twenty-five public university in the country, improve the College of Public Health and College of Medicine, and increase the popularity of its football program.
In 1993, Borkowski was elected to be Appalachian State University's (ASU) fifth chancellor. During his tenure he expanded the University's outreach program. Holmes Convocation Center, a science building, was built during his time at Appalachian State. Though the university had improvements, there was controversy to some of the actions executed by Borkowski. Between 1995 and 1997, many rapes occurred on university campus. Some think that the rapes were not handled properly and the rapists were not given proper consequences. Borkowski was blamed. At USF in 1992, he also questioned to why he did not take action when a basketball star athlete reportedly rapes and/or sexually assaulted six women over the course of many months. The athlete did not receive any disciplinary punishment and was still eligible to play in 1989 and 1990. In 2002, he used a designated area on campus to create a "free-speech-zone", which was created in response to students protesting American foreign policy. Foundation for Individual Rights in Education (FIRE), a civil liberties group, questioned the constitutionality of the zone. In response to FIRE, ASU only limited it's "free-speech-zone" acts. ASU He resigned as chancellor in June 2003.
Borkowski met his wife, Kay, at IU. They married in 1959.