I believe that what Davis meant by this is that Turner was never just another slave.  He was never (Davis is claiming) someone who was willing to simply go along with his enslavement.  Instead, he was a proud man who fought back against his oppression.  In this sense, he was "always alive."

Turner claimed to have always known that he was special.  He took pride in having learned how to read and in having done so quite easily.  He said that his parents had told him that he was destined for greatness and he, himself, believed that.  Later on, of course, he felt that God was talking directly to him.  This was a man who was not just like everyone else.

What Davis is saying is that Turner was a special man.  He was more alive than everyone else because he had such a burning sense of his own special nature.  He was always "alive" in the sense that he was always ambitious and always striving.

This is, I believe, what Davis means when he says this about Turner.