You didn't say where you are; makes a big difference.
Thyme is perennial and cold hardy. It will be done for the year when frost hits (assuming you live somewhere with cold winter), but will come back next year.
Basil is annual. It will keep growing through the season, if you keep cutting it back and not letting it go to seed. Once it has flowered and gone to seed it is done. If kept cut back, it is done when frost hits. In frost free areas it is done at the end of the year any way, because it gets woody and less productive.
I have two methods I use for drying herbs. You can cut long sprigs (cutting just above a leaf node and not cutting down in to woody stems), bunch them and hang upside down in cool dark dry place. I think it helps to hang them inside a paper bag - the paper absorbs humidity and keeps the herbs drier and dark. Once they are crumbly dry, strip the leaves off the stem and store them in glass jars away from sunlight.
Or spread the herbs out on a baking sheet so there is just one layer. Turn the oven on the lowest setting and leave it on for no more than ten minutes. Turn the oven off without opening the door. Leave the herbs in the oven for a few hours. If they aren't fully dry, turn the oven on low for ten more minutes and off. Leave the herbs in the closed oven overnight. They should be dry by the next day. Store as above.
Dried herbs can be shipped in paper envelopes. You would want a small envelope packed inside a heavy mailing envelope.