The Greeks and Romans did not have germ theory as we know, but they did know that boiling water purified it. The best attestation for this is found in Hippocrates, as mentioned in the Wikipedia article on water filters:
Hippocrates believed that water had to be clean and pure. Rainwater was the best water, but had to be boiled and strained before drinking to get rid of the "bad smell" and to avoid hoarseness of the voice. He designed a crude water filter to “purify” the water he used for his patients. Later known as the “Hippocratic sleeve,” this filter was a cloth bag through which water could be poured after being boiled.
The actual quote from Hippocrates is from De Aere Aquis et Locis 8:
Such waters are naturally the best. But they need to be boiled and purified from foulness if they are not to have a bad smell, and give sore throat, coughs and hoarseness to those who drink them.
That unboiled water in general can be unsafe can also be inferred from the strictures of Celsus on treating sick patients (3.23.7), who says that:
no water should be used for drinking unless it has been boiled.
Aristotle in his Meteorology mentions filtering seawater to make it drinkable, and describes well the purifying effect of boiling, but doesn't connect the two explicitly (since the book's concern is with meteorological phenomena, not sanitation.
I also see quite a few articles and books on sanitation/disinfection mention Aristotle mentioning disinfecting water explicitly, but none of them have a direct citation. They might have confused him with Hippocrates, or perhaps thought the evidence was more explicitly laid out than it really is.