There are some pre-scientific ideas about the cause of disease that, if they were consistently acted upon, could mitigate disease.
For instance, the Bad Smell theory of disease has been attested in Roman writings as early as the first century AD. If it gets taken seriously on a society-wide scale, it implies that human and animal wastes, rotten food, corpses, and dead animals should be promptly buried, burned, or carted away from populated areas. Do this, and you remove a lot of contamination-based disease vectors (particularly for things like cholera and dysentery) as well as drastically reducing the food supply for rats and other vermin.
I also remember reading about field sanitation practices in classical armies: if memory serves, Greeks, Romans, and Persians generally understood the importance of burning or burying your wastes and your dead bodies away from and downstream of your army's camp. Extend these principles to cities. I know there were attempts at urban sewer systems in Roman times, but I'm not sure how effective they were from a sanitation perspective or whether they were still widely used in the ERE by Justinian's time. For the Black Death, though, urban sanitation in Western Europe consisted largely of a combination of "I hope it rains soon" and "Maybe a dog will eat it".
Classical peoples also had an inkling of the principles of contagion, at least to the extent that disease bred disease and close contact with an infected person put you at heightened risk of infection. I know there were attempts at quarantines (both locally upon plague ships and houses of infected individuals, and at a city-wide level) in many classical and medieval plagues, but there's plenty of room for improvement. For instance, the concept of an incubation period (if it occurs to people and gets taken seriously) implies a need for ring quarantines (i.e. quarantine not just the site of infection, but also a ring around it to capture most of the as-yet asymptomatic incubators), and for a quarantine period for travelers coming from the direction of infected areas even if the travelers themselves aren't showing any signs of illness yet.
There's also a correlation of plagues tending to hit during or shortly after a period of war or famine: these break down institutions that could enforce effective quarantines, and they tend to cause society-wide shortages (particularly malnutrition) that lead to a big chunk of the population having weaker-than-usual immune systems. If you want to mitigate a particular plague, improve nutrition and social order in the years leading up to it.