Born in Jackson, Mississippi, Alsup received a Bachelor of Science degree in mathematics from Mississippi State University in 1967, a Juris Doctor from Harvard Law School in 1971, and a Master of Public Policy from Harvard University's John F. Kennedy School of Government in 1971.
He was a law clerk to Justice William O. Douglas of the Supreme Court of the United States from 1971 to 1972. Alsup was in private practice in San Francisco, California from 1972 to 1978, and was then an Assistant to the United States Solicitor General in the United States Department of Justice from 1978 to 1980. He returned to his private practice in San Francisco from 1980 to 1998 with Morrison & Foerster, when he briefly served as a special counsel in the Antitrust Division of the Department of Justice in 1998. He was again in private practice in San Francisco from 1998 to 1999.
On March 24, 1999, Alsup was nominated by President Bill Clinton to a seat on the United States District Court for the Northern District of California vacated by Thelton Henderson. Alsup was confirmed by the United States Senate on July 30, 1999, and received his commission on August 17, 1999.
Alsup was the presiding judge over Oracle America, Inc. v. Google, Inc., where he notably was able to comment on issues relating to coding and programming languages, specifically Java. He learned the Java programming language solely for the purpose of being able to understand the case more clearly. However, the Federal Circuit overturned his rejection of the copyrightability of Java API.
Alsup was also the presiding judge in what is believed to be the first trial against the U.S. no-fly policy, which is a list of people who cannot use commercial aircraft in the United States. Regarding the removal of people incorrectly included in the list, he ruled that, "he government's administrative remedies fall short of such relief and do not supply sufficient due process."