I have a variety of herbs and this mildew (or whatever it is) has been spreading over and over again:

I threw out the infected plants after culling what I could and got new plants. They are in separate rooms and everything is getting infected again! Since I don't know what it is, I can't treat it. It looks like it might be downy mildew? I do find little dots (spores, I guess) on the underside of some (but not all) infected leaves. Could someone please help me identify this disease and how to treat it?

If it helps any: All of my herbs are on indoor window sills (I have zero outdoor space) getting southwest exposure and I live in NYC. I got the herbs within the past month and it's been in the 70s/80s. Humidity from 30-60%. They get breeze throughout the day. I water the soil, not the plant, so the leaves get no extra moisture.

Any ideas? Please help!