This is most likely a fungal leafspot disease affecting your asters. Although asters are native, they are susceptible to some diseases, including botrytis and septoria. Sanitation is key in controlling the diseases. Thoroughly clean up and destroy dead tissue as it appears, and remove all aboveground plant parts over winter. Provide good air circulation around plants so that they dry quickly. Avoid applying nitrogen fertilizer to asters---it makes them more susceptible to disease. If the problem persists, you may need to apply preventive fungicides early in the season when conditions are wet.