The airport security would be brought in to investigate the situation. The couple would be detained while the incident is investigated. The police would also be called in, and the couple could be criminally charged with assault and other offenses depending on what happened.
The airline personnel would provide medical attention for the baby, and provide care and accommodations if needed. If the couple is deemed a safety hazard, the airline would ban them from the aircraft and update their no-fly lists.
The airline would also investigate the incident and collect evidence of what happened, and file a complaint and any necessary documentation. If after the investigation, the airline is able to prove that the couple endangered the safety of the aircraft, the airline could take legal action against them.