Alexander Acosta's parents were Cuban immigrants who fled Cuba as refugees, settling in the United States. They raised their only son, Rene Alexander Acosta, in Miami, Florida, where he was born and grew up. His father worked as a civilian employee for the Department of Defense, contributing to the family's stability in their new country. As first-generation immigrants, they represented the challenges and aspirations of many Cuban exiles during the mid-20th century wave of migration following the Cuban Revolution. Their background as refugees underscores the immigrant roots that shaped Acosta's upbringing and emphasis on education as the first in his family to attend college.