It could be powdery mildew or mold. Here is a old post with useful comments.

Yes, cut the flowers off!

Indoor herbs almost always end up weak and diseased. Indoors they get insufficient light and there is no weather or predators to keep down pests. Do you have anywhere outside you could put them?

Do the pots have drainage holes?

Thanks for the suggestions. I'll try putting them outside for a bit. It's nothing to do with light then? I could move them somewhere else inside with more light but if that's not the problem then I guess outside is the way to go. We don't really have a garden, but I'll find somewhere.

Powdery mildew. The primary problem is as floral says....it's growing inside. Herbs love the sunlight, breezes, rainfall, and humidity found outside. Your plants are seriously unhappy.

Mind you, herbs can be made to languish on a sunny windowsill for a season, but they need excellent light, a high quality, porous potting mix, and attention to nutrition.

ok, thanks rhizo! i've put it outside for now, hopefully that'll help!

As I said, "indoors they get insufficient light". If you have nowhere outside at least put them in as bright a spot as possible.

Don't introduce the plants to full sun right away. Allow for several days of acclimation or they will be fried to a crisp. Also, the PM looks serious and sunlight won't fix it.

Even 'full' sun is pretty weak here at this time of year and I wouldn't worry too much about acclimatisation. I also believe that the PM would clear up outside because it prefers dry conditions, low light and poor air circulation. Especially if you trim off some of the worst affected stems.

I am wondering whether you got these plants from the veg section of a supermarket rather than a garden centre? If so they are grown fast and sappy for consumption in the kitchen rather than more robustly for planting in the garden and they don't usually last long.

I've cut off all the affected parts of the plant, given it a wash and put it outside for now. I'll see how it gets on. Yes, I think this was probably a supermarket plant but I've had it for a year or so and gotten some good use from it. Hopefully it'll revive itself outside. Thanks for all the suggestions!