As I’ve mentioned, blackberries require very little maintenance to be productive. However, there are things you can do to help them reach their full potential. First, I like to give them support by constraining them in a wired fence. You don’t need much here, but it will prevent the canes from falling over on the ground, making them less prone to disease.
You also need to remove the dead floricanes once they are done bearing fruit. They are unattractive, but they are also a source of disease. Cut them off at the ground and remove them from the area.
Other than that, all they need is a small amount of a balanced fertilizer (I use 10-10-10) and some supplemental water when you’re going through a dry spell.