It really does look (and sound) like rust fungus, but there may be other issues. For example, what did you amend your clay soil with (and we're assuming you planted them in the ground and did not amend the soil and put them in pots)? It sometimes takes quite a bit of amendment to make a clay soil drain well enough for plants like this, that like very good drainage. If the plants are not healthy due to root issues from too much soil moisture and not enough oxygen available to the roots (clay soils can also be very compact and air does not move thru these soils), they will be more susceptible to diseases, even those they don't usually get (but which can infect them). We recommend you look closely at the leaves, especially the underside for the fungal rust spores (orange and/or dark brown to black) and if you find this we recommend you get a fungicidal product (neem oil may be appropriate) and follow the label directions before making application(s). Full sun is correct for these plants and they should be watered when soil moisture is almost depleted in a well drained soil. This may take longer in a clay soil, but in either case, we recommend watering deeply less frequently, rather than smaller amounts of water more frequently. We also suggest you take a couple of stem & leaf samples (sealed in a bag) to your local garden center and try to get a second opinion; provide as much info to them as possible about the care given to these plants. We hope this helps resolve the issue, but if all else fails, you could take another sample (also sealed in a bag) to your local Dept of Agriculture Office and they will usually send the sample to the county plant pathologist and report back to you for free.
This small pot actually has several basil plants in it. There are several reasons it could be drooping, and often with plants what we see isn't the response to just one condition but a combination of conditions. Here is what may be going on:
1. Yes, you are correct that Basil likes it on the warm side. So cold from the window (especially the temps in your area on Wednesday night/Thurs AM!) can make them droop.
2. Lack of good light will cause the older leaves to yellow and fall off. So if you bought this plant to use in cooking, pick the largest, older leaves first to use before they yellow. You could put a regular lamp with a "gro light" in it near the plant and keep it on for 12 hours a day - that would help. Such full-spectrum lights need to be very close to the plant to do any good.
3. Over watering and under watering cause the same symptoms: drooping leaves. Water the plant really well and then wait until the soil is dry, but before the plant starts to wilt, to water well again.
4. You might want to put this group of plants in a larger pot asap - that will help even out your watering schedule and encourage the plants to grow. Since you're a new gardener, here's something you should know about plants: what happens in the soil is reflected up above. So a plant that has room to grow more roots will produce more growth up on top. But a plant that is root-bound, and doesn't have enough space for root development, won't produce more stems and leaves.
5. Fertilize with a general fertilizer used according to directions after you've watered the plant recently. (Another tip: never fertilize a thirsty plant.)