Think about it this way: Colonel Klink was a member of the German aristocracy (a Junker), even if his name lacked the wealth that normally comes with nobility. He was also a member of the regular Germany military. Both groups hated Hitler and the Nazis and even tried to assassinate the former many times. I mean, hell, Sgt. Schultz is depicted as equally dumb, but he at least had some knowledge that Hogan and Co. were up to something. If he knew, then Klink had to too, right?
Speaking of Schultz, it's weird how he's portrayed as an idiot, but yet, he ran the largest and most successful toy company in Germany before his factory was converted to war use and he was recalled to the military. How do you run a successful company and be as dumb as Schultz is depicted on the show?
Unfortunately, as The A.V. Club points out, Hogan's Heroes was never given a proper series finale, so we have no idea how things ended. I'd like to think though that if it did have a finale, it would be revealed that Klink and Schultz knew. It would also have them helping the POWs make their escape from Stalag 13 (no idea if the camp was in the danger from the Red Army or not) with the two Germans escaping with them as Hogan's "prisoners".