Not a historian, but a history fan and I am familiar with Koine Greek - the language of the New Testament.

While the account doesn't specify for whom the jailer worked (simply ho desmophulax or "the jailer"), it is likely safe to assume he is a Roman soldier; Philippi was a Roman colony (Acts 16:12), and Roman soldiers from cohortes urbanae (Urban Corps) were typically responsible for jailing duties (Tacitus Ann. II.) because Roman Provincial Governors had no police force at their disposal, policing in the colonia was performed by Roman legionnaires from the local fort (Burton, G. (1988). The Roman World (J. Wacher ed.). pp. 424-426.).

While I can't find any primary sources that illustrate a jailer's punishment specifically (feel free to help me out here), there are many colorful primary accounts of punishment befitting a Roman soldier who has violated his sacramentum, or oath of duty. Polybius in his "Histories" (Volume III, Chapter 39) describes punishment ranging from beatings, ration reduction and additional duties for minor infractions, to fustuarium for special occasions.

Fustuarium was a method of capital punishment pronounced upon those who participated in exceptional acts of cowardice, abandonment of post, or dereliction of duty. This often included public humiliation, a severe flogging, and finally, death by clubbing or stoning. If a Roman soldier had allowed every single prisoner in his keep to escape - especially with the excuse that a magic earthquake had shaken the door free - his Commanding Officer would have had a field day with him. It would end with his violent, long, brutal death, and a social stigma attached to his household that would likely take generations to erase. Suicide was a much better option for him.

EDIT: Clarity

EDIT2: There is no mention of this paramilitary branch being stationed outside of Rome, Lugdunum, or Carthage. It seems it was a specialized branch for riot control. (Le Bohec, Yann. (1994). The Imperial Roman Army. BT Batsford Lt. pp. 100. Link

In the Satyricon of Petronius (paragraph 111-112) a soldier guarding the crosses to forbade proper burials to bandits is distracted by the mourning of a widow crying days after days on the corpse of his husband. He try to consolate her and then began a relationship with the widow. Relatives of one of the bandits then recover his body. When the soldier sees the empty cross he tells immediatly to the widow that this tomb will see the death of two men. In the end the couple uses the corpse of the deceased husband and hang it to the cross to save the life of the soldier.

Apparently he expected to be executed, even if he was just guarding corpses.

I know I've been taught that a Jailer would have been responsible for the penalty the prisoners would have faced. In this case, that would be death. Unfortunately, while I'm finding a lot of people stating the same, I'm struggling to dig up any primary source.