Plants of the Aster family produce tons and tons of seed, but they are notorious for it mostly being nearly all chaff. Thank goodness, think of the problems we would have if it were all viable!!! If you remove the old flower heads before they open to fluff, you'll catch a lot of it before it can spread.
Not sure what size aster your looking for, but I've grown my low gowning Snow Flurry Aster for 24 years, never deadhead, and never found a seedling.