Sounds like either root rot or foot rot. Both are fungi which is harbored in the soil as a result of either poor air circulation, poor drainage, overwatering/excessive rain, or all three.
You
could try a cornmeal spray or dusting and see if this helps to halt its spread. But when this happened to me I removed all the browned foliage and some even further up. Then I cut the plants back 1/3, removed whatever mulch that was below and around them and even removed the top couple inches or so of soil.
If the plants had gotten too wilty at the top as you mentioned some of yours are, then I would've cut them back by 1/2. If they were just too far gone, I would've yanked them or just cut them down to the ground and chalk that season's bloom up for lost. Once they're that infected w/the fungus, they really wouldn't have bloomed all that much anyway. Or at all.
If it had happened the following season, I would've removed the plants from that spot altogether because that would have meant that the fungus was still present in the soil. I would've just planted something else there that wasn't as susceptible as the asters and gotten new asters to plant in a different location. Fortunately, I lucked out the following seasons and it never happened again.
Hope some of my experiences w/this can be of some help to you.