Simon Lewis, 33, from Trowbridge, was killed in the crash on Lamby Way on New Year's Eve.
His pregnant wife had an emergency Caesarean Section on Sunday, but the infant died the same day.
Kyle Kennedy from Rumney was remanded in custody at Cardiff Magistrates' Court on Friday.
In addition to two charges of causing death by dangerous driving, the defendant was also accused of driving without insurance, failing to give a blood sample, driving while disqualified and aggravated vehicle taking without consent.
Mr Lewis was the driver of a Daihatsu Sirion carrying his wife Amanda and their three-year-old daughter when it was involved in the collision with a Peugeot 307, belonging to Mr Kennedy's girlfriend.
The Lewis's son was born three months premature when a decision was taken to carry out the emergency Caesarean Section, amid concerns the baby was in distress.
A 29-year-old man has appeared in court accused of causing the deaths of a baby boy and his father in Cardiff by dangerous driving.