It was not at all significant to the beginning of the Cold War, because the Cold War was already happening in 1961. Most historians date the Cold War back to 1947 and the Truman Doctrine, ir the Berlin Blockade a year later. Also when the Wall was build it wasn't seen as as significant as we would today. The German border had already been fortified and after the Allied made sure their access to the Western Part of the City would not be compromised again they did not intervene.

While it had impact on the German society on both sides and was seen as symbol of isolation from its inception, on the Grand Scale of the Cold War the building of the wall itself is more a footnote.