Answering this from work so I'll have to compile more sources when I get home.

Generally, reigning in the fire hazards of these structures was a slow process and usually took major events like great conflagrations (specifically the Great Boston fire of 1872, or Salem's conflagration of 1914 come to mind) to force cities and towns to adopt more strict fire codes. Of particular note is a quote from the Salem Rebuilding Commission Report, which stated that prior to the fire, "It must be borne in mind, too, that the City before the fire was fast on the way to become a city of ugly and unsafe wooden three and four deckers." Builders, on the other hand, mostly did as they pleased, only developing safer building practices as the risks became more obvious.

The greatest risk of fire from triple-deckers came not only from their wooden materials, but also because of their use of balloon framing, which was a new way of wood framing adopted during the mid 19th century. Instead of using substantial timber frames that would spatially divide floors through their framing, balloon framed houses would have studs going top to bottom with no solid framing to block air movement at the floor level. This allowed fires to travel quickly from floor to floor. As builders adapted to the new design, they eventually realized that fireblocking was necessary, and would install brick or wood obstructions at floor levels to stop this chimney like behavior from happening within the home.

While individual builders might be concerned with how fire would spread within a structure, cities were more focused on stopping the spread of fire from one building to the next. Because triple-deckers were not connected, spread of fire through conduction was not the primary concern (like with row houses which would use brick firewalls to reduce the spread of fire). Instead, convection, radiation, but most importantly, wind-driven embers, were factors to consider in the risk of fire spreading in dense urban areas populated with triple deckers. Most of these buildings were not only sided with wooden shingles, but roofed with them as well. When firefighters had to stop fires, water pressure was often a crucial factor, and if there was not enough pressure, hoses could not reach rooftops to saturate them. This would leave wooden rooftops dry and exposed to floating embers.

To address this, cities like Boston and Salem quickly instituted fire codes that required fireproof rooftops made of asphalt, tar or gravel. The Salem Rebuilding Commission Report required "Every new building of wood has a fireproof roof and gutters, so that sparks and embers falling from above cannot set them on fire."

From a building design and construction perspective, these were the two main issues to address to reduce the risk of conflagrations, especially with concern to triple-deckers. Taller buildings with wide open elevator and ventilation shafts had to have fire-suppression systems installed and shafts closed off to stop fires from shooting up them like chimneys. Beyond that, most fire safety improvements came from technologies that allowed for greater fireman response times and water hose pressure.

Sources:

Urban Conflagrations in the United States by William M. Shields, Ph.D http://www.tvsfpe.org/_images/conflagrations.pdf

Report of Salem Rebuilding Commission 1917 - https://digitalcommons.salemstate.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&httpsredir=1&article=1002&context=fire_documents

Boston (Mass.). Commissioners to Investigate the Great Fire Rockwell & Churchill, printers, 1873 - Boston (Mass.)

Garvin, J. L. (2001). A building history of northern New England. Hanover: University Press of New England