There appear to be overlapping issues: it looks like there is some powdery mildew present (a very common fungal disease for this time of year), and probably Chrysanthemum Lace Bugs causing damage that is leading to premature leaf aging and shedding from the bottom up. The lace bugs begin to feed earlier in the year (they overwinter as adults that take shelter in leaf litter), but only by late summer or so does damage tend to accumulate to the point of drastic symptoms. You can look for them by examining the leaves (not the brown ones, but live leaves further up the stem) on both the upper and lower surfaces. Most lace bugs feed on the underside of leaves, but Chrysanthemum Lace Bugs in particular can be on either surface.

If you don't find lace bugs, they may have already been eaten by predators, and no intervention is needed. If you do find some, they can be hosed off of the plant with a strong spray of plain water from a garden hose. If a pesticide becomes necessary to knock-down a high population, you could consider spraying with either horticultural oil or insecticidal soap. Do not apply either while temperatures are above 85 degrees, or it will risk plant tissue damage. Follow all product label directions for use, and do not spray open flowers. These low-toxicity insecticides only work when they directly contact the pest, so if it's too difficult to get good contact with the lace bugs (or if you don't see any around), then it won't work well.

Fortunately, despite creating an eyesore, these pests don't tend to cause enough damage to the plant to be concerned about, and the aster should resprout normally next year as long as its roots aren't affected by an unrelated problem. If you'd rather not spray and want to just improve the appearance of the plant for now, you can use a gloved hand to brush off the leaves (gently close your hand around the stem and run it from the bottom up, over the area of dead foliage, and it should knock/push most of them off).

Miri