The "insult" which some people see in this phrase does not stem from the ACLU element, but rather the card-carrying element. It's a holdover from the McCarthy era, when "card-carrying" expanded to "card-carrying member of the Communist Party". Many people with long memories and/or short educations still see the one and assume the other.
This phrase was used successfully by George Bush in 1988 to cast discredit upon Michael Dukakis during the presidential campaign for precisely this reason. Ironically, the ACLU was defending Oliver North at the time.
(Update: 16 Aug 2000) Mea culpa. I had a brain fart when I originally wrote the second paragraph, and I attributed some quotes to Bush that were actually used by George Smathers against his opponent, Claude Pepper, in the 1950 Democratic Senate campaign in Florida. The two bits of information were listed consecutively in an article in Uncle John's Bathroom Reader, my original source, and my memory blurred the boundary between the two. Oops.