Were Basil of Caesarea and Jerome alive at the same time?
Yes. Basil of Caesarea and Jerome were alive at the same time, from c. AD 345 to AD 379.
Basil of Caesarea
4th-century Christian bishop, theologian, and saint
Born AD 329 · died AD 379, aged 49 or 50
Jerome
4th and 5th-century Catholic and Eastern Orthodox saint and Doctor of the Church
Born c. AD 345 · died c. AD 420