There does not seem to be much, if any, information about adverse affects fungicides may have on insects, although some are reported to aid in control of some of the insects that can carry some plant diseases. It is also known that some fungicides do have an adverse health affect on bees, so it would not be much of a stretch to think fungicides could be harmful to other insects.
There must be a good reason why many of the labels state the product should not be applied when insects are active.

Do some online searching to learn about an aster problem moving south known as "aster yellows". Sounds like the problem. No treatment to date and the disease also affects other plant families. Every gardener should learn to recognize the symptoms of this disease, the plant types it affects and know to pull and destroy infected plants PDQ should it appear. Be alert to this one as the problem is spreading.

I use Copper Fungicide on my grapes, which is supposedly the best "organic" stuff you can buy in stores. However though, right on the back you have "Danger: Highly Toxic, don't inhale, get on skin...". This just goes to support the fact that Organic doesn't necessarily mean better, that stuff is no different than a pesticide.

Asters in my region develop "drying" lower leaves just about the time the plants should bloom. When I look very close, I find dozens of small caterpillars hidden among the debris.

If that's what's happening with yours, the pests are safe from any spray because they are hidden among the debris.

There are quite a few organic fungicides available that won't harm insects but they may or may not target your specific type of fungus issue.
I use Actinovate as a preventative for downy mildew, early blight, powdery mildew and grey mold. Oxidate also works well when a fungus is there already and damaging plant tissue. You wouldn't want to spray it directly on a beneficial insect because it's food grade hydrogen peroxide, but once applied should not harm any insect.
Compost tea is another option as a preventative.

Hope this helps. If it's aster yellows, i've heard that's difficult to eradicate.
-Mark

Thanks to everyone for the responses.

Nandina, I've been doing some research on diseases of asters, including aster yellows, but nothing seems to fit exactly. My best guess is either aster yellows or botrytis blight. I'm leaning toward botrytis blight, since the plants went downhill so quickly and botrytis likes wet conditions. We've had a very wet summer here in central NC (almost 2x normal amounts of rain in July & August).

I'll keep up the online research, but from my reading so far it seems like the best way to manage fungal diseases without harming pollinators is to encourage healthy bacteria through things like compost teas.