Not sure what value this has, but the NFPA firefighter 1 textbook seems to only treat the Roman origins of firefighting, then jumps to the foundation of the US volunteer fire companies. Either way, the relevant paragraph on page 10 of the textbook is:

Since prehistoric times, controlled fire has been a source of comfort and warmth, but uncontrolled fire has brought death and destruction. Historic accounts from the ancient Roman Empire describe community efforts to suppress uncontrolled fire. In 24 BC, the Roman emperor Augustus Caesar created what is probably the first fire department. Called the Familia Publica, it was composed of approximately 600 slaves who were stationed around the city and charged with watching for, and fighting fires. Of course, because the Familia Publica consisted of slaves, these conscripts had little interest in preserving the homes of their masters and little desire to take risks, so fires continued to be a problem.

In about 60 AD, under the emperor Nero, the Corps of Vigiles was established as the Roman Empire's fire protectors. This group of 7,000 free men was responsible for firefighting, fire prevention, and building inspections. The Corps of Vigiles adopted the formal rank structure of the Roman military, which continues to be used by today's fire departments.

The first rudimentary hand-pumped fire engines were developed in the 18th century. Before this, I remember a story about local ordinances in Philadelphia requiring every household to maintain a bucket of water in front of their home for the fire brigades. In the event of fire, the stored water would be used as the basis for a bucket brigade. One group of people would be hauling whatever water is available, and another would be attacking the burning structure from the outside with pike poles - to remove the burning pieces and make them easier to extinguish. The pike pole was so useful that it's one of the main tools still used today when overhauling the fire scene - works great for ripping down ceilings and breaking into wall spaces.

The other major risk management technique used for fire hazards in premodern times were simple construction codes. Think of London's transformation after their great fire, it represents the larger shift in historical building construction towards noncombustible materials such as stone and brick in dense urban areas. If I recall correctly, Rome went through such a building code change too around the time that the Corps of Vigiles was established - and it's likely that they used similar tactics to what prevailed elsewhere before the invention of pumpers.

(Edit: And there was no putting out mass fires back then. When both Rome and London burned, there was nothing any human being could have done to prevent it - even with modern equipment. Just look up the Asbury Park NJ boardwalk fire from this past year. After a certain point, the line between structural and wildland firefighting is indistinguishable.)