Apollodorus of Damascus was Emperor Trajan's architect. He is given credit for the Baths of Trajan and the Forum of Trajan, among other buildings.

As the name indicates, the Baths of Trajan were built by the emperor Trajan.

The Roman Emperor Trajan spent the gold extracted from the conquest of Dacia on many buildings in Rome. One area Trajan wished to improve was the public baths. Reportedly, he had the architect Apollodorus of Damascus design a huge complex of public baths. Citizens could enjoy hot and cold baths. The baths were a great way to socialize in ancient Rome.

The name of the Roman baths was thermae. Only in the city of Rome, where there were many baths, there were distinctive names for baths: the Baths of Agrippa, the Baths of Nero, the Thermae Etrusci, the Baths of Titus, the Baths of Domitian, the Baths of Trajan, the Baths of Caracalla and the Baths of Diocletian.

Thermae Etrusci is a term coined by historians. They were commissioned by Claudius Etruscus, a freedman at the court of the emperor Claudius who became the head of the imperial financial administration.

Depends which ones to which you're referring:

Baths of Caracalla were constructed around 215 A.D.

Baths of Trajan were constructed around 105 A.D.

Baths of Diocletian were constructed around 300 A.D.

Those were the main ones, and they were built during the reigns of the emperors I have typed above.