When I was a kid, we picked the blackberries and never did anything else to them at all.
But when I bought the 10 acres the end of 1994, there was a row of blackberries in the back yard. So I got some literature from my county extension agent on caring for blackberries. The canes that grow this year will produce next year and then die. New canes will be growing next year for the following year's crop. The old dead canes, if left in place, can theoretically, at least, be more susceptible to disease and insects. Therefore, the recommendation is to get rid of the old canes when they quit producing for the year. In my part of the country that was sometime in June. Ideally, you'd cut out the old canes and leave those 6' canes that had already grown this year, but since mine were not thornless, that could be a slow and unpleasant task. The alternative recommendation is to just mow everything down when they quit producing. Then apply fertilizer and water to grow new canes for the next year.
Now that was a recommendation more to my liking. The first year, I cut them off just above ground level with the metal circle saw blade on the string trimmer while my brother pulled them out of the way with a rake. Too much work, and then I had all the old bushes to burn. So from then on, when they quit producing, I mowed them down with the brush hog, then followed that with the finish mower. That cut them, mulched, and scattered the clippings.
Now the canes that came back and produced the next year never got as tall as what I cut down in June, so maybe I'd have had more berries is I'd only cut the old canes out, but we still had all we needed every year for cobblers and making blackberry jelly. My little blackberry patch was approximately 4' wide and 50' long. They would try to spread more, but I just kept anything more mowed. The below picture was taken on April 16, 2002.