What to do with herb garden boxes/trays for winter?

What to do with herb garden boxes/trays for winter?

Hey there! I have two large 1' X 2' boxes of growing herb plants that I've been using in cooking all summer that just got assaulted by the first snow of my area (Minneapolis). The basil was pretty much obliterated, but I still have a bunch of oregano, parsley, rosemary, french tarragon, and mint. What do people do with these kind of plants for winter typically? I could bring them in, but for some reason they seem more likely to bring bugs/mold in with them (maybe it's the amount of dirt surface area) and available space for trays their size is pretty limited. Is there a good way to dry herbs like this for later use? Oh, and if any of you have a recipe which calls for a lot of green herbs, I wouldn't mind hearing some!
Pictures of a red sunflower.

Pictures of a red sunflower.

Plant ID, Fellow Plant Nerdies

Plant ID, Fellow Plant Nerdies

Redecorated and Rearranged Living Room aboard NAS Meridian

Redecorated and Rearranged Living Room aboard NAS Meridian

My front lawn

My front lawn

Weed or marigold?

Weed or marigold?

Bug ID and Lily ID

Bug ID and Lily ID