Unless you consider them very unsightly right now, there's no reason not to let them stay green as long as possible. You can cut them back after they have frozen and died, or even cut them back in spring.

They will naturally die down all the way to the ground when they do go dormant. The stems should basically fall out on their own or you can cut them down to 1-2 inches.

They may be established well enough survive winter whether you cut them or not, but you'll certainly improve your chances even more if you leave them green as long as possible. September-planted mums sometimes make it and sometimes they don't. But the green leaves will allow the roots to continue getting deeper.

I tend to cut them both down, the asters because they tend to get mildew and I want to reduce the amount of disease that winters over. My particular mums, which are a totally winter hardy, late-blooming variety, have pretty tough stems, so if they stop blooming before the snow buries them I cut them back to remove the dead stick look. Most often, however, they disappear beneath the snow before they stop blooming and they get trimmed back when they emerge in the spring.