Klaus batted left-handed and threw right-handed, and was listed as 5 feet 9 inches (1.75 m) tall and 160 pounds (73 kg), during his playing days.

After nine games played and seven at bats without a hit during brief trials for the 1952–53 Braves, Klaus was included in one of the most important trades of the early 1950s, when Milwaukee sent him to the New York Giants on February 1, 1954, with pitchers Johnny Antonelli and Don Liddle and catcher Ebba St. Claire (plus US$50,000) for erstwhile Giants' hero Bobby Thomson and catcher Sam Calderone.

The deal came out heavily in the Giants' favor, with Antonelli pitching them to the 1954 National League (NL) pennant and world championship.

Klaus never appeared in a Giants' uniform; he was the all-star shortstop in the Triple-A American Association (as a Minneapolis Miller), hitting 21 home runs (HR), and was acquired by the Red Sox during the 1954–55 offseason.

After his active career, Klaus managed Minor League Baseball (MiLB) affiliates of the Senators and Oakland Athletics.