According to William Manchester's biography of MacArthur ("American Caesar"), there were various reasons. MacArthur was committed to a policy of genuine freedom of speech and toleration of dissent (up to a limit, but it was a pretty wide limit); American political reforms had pre-empted many of the Communists' talking points; and many Japanese rejected the Communists as a result of Stalin's refusal to release Japanese prisoners of war after the end of the war.

MacArthur did move to prevent a Communist-led general strike called for May Day, 1946, and apparently his framing of it as an irresponsible and dangerous action against a fragile recovering society succeeded, and resulted in decreased support for the Communists. So he apparently didn't see any need to ban the party because in his assessment it wasn't a problem (except in the one case of the general strike).