Hey newgirl:
I don't know what part of the state you are in but down here in Snowflake, (Va.), we can usually coast until the tail end of October with Basil, Sage, etc. Unfortunately the Basil has probably started blooming on you and you have missed the peak time to make pesto. (before the bloom stage) but it will still taste better than store bought. Another thing to consider is to cut it coarse (less bruise) and freeze it in your ice trays and then put in storage bags (read ziplock) and freeze same until you wish to add it to soups stews whatever. If it has made seed.....save it for next season and use quicker. Welcome to the wonderful world of gardening....never enough time.

Hi newgirl and welcome to the forum! Yes, compost everything as it dies back except those things that are diseased. Those should go to the landfill unless you are seriously into composting and maintain a hot enough pile to kill off the diseases.

There's no such thing as a silly question if you need an answer. I hope by now you've had a chance to harvest the basil. Like mmstiner said, it will be better than anything you buy in those little jars at the store! I sometimes let it go to seed so I can avoid having to buy seeds the following year and then harvest the leaves to dry for winter use. Yes, it's better when younger, but who cares when it's so good anyway!

Good gardening!
Sandy