It's debated whether Truman ever was a member of the KKK. The main proponent of this narrative was Spencer Salisbury, who, along with Edgar Hinde, convinced Truman to apply for membership in 1924, when he was running for reelction as a county judge in Missouri. The theory being that his democratic opponents were backed by the Klan and so he needed Klan support to mitigate that. This was, of course, when the Klan was at the apex of its political influence in the US.

Truman and Salisbury would later have a falling out over business differences, and Salisbury would then claim Truman was a Klansman. Hinde and Truman himself denied his ever actually joining, saying he applied, but backed out when he was told to disavow Jews and, more importantly, Catholics. Truman's major political benefactor was the Pendergast family, who were themselves Catholic. You can read an interview transcript with Hinde, where he clearly denies Truman joining the KKK, here.

As for impact, it's safe to say whatever relationship he had with the KKK had very little impact in the long term. Truman would famously integrate the military in 1948, something its unlikely he would have done had he harbored racist views typical of the Klan.