Blackberries produce fruit on second year wood. After that the old vine is nothing but a drain on water and nutrients. All the vines that grow after harvest are next years producing vines. Trim everything that had fruit on it off leave all the new stuff.
Thanks. I guess I should do this as soon as they stop producing so I can remember which is which. This time I just cut all the living main stalks off at about 3 feet and the lower branches off at about 1 foot from the main stalk. I should know next time, but I'll probably ask you again next year.