For me, the 1960's is a hit or miss decade for comedies. Some I've loved, others, not so much. Unfortunately, That Touch of Mink is part of the latter.
The only thing I enjoyed about this film was Cary Grant. He (as usual) was decent as the suave, debonair gentleman that he often played, but I wouldn't consider it one of his best roles by any means. Doris Day, who I've never cared for, delivers what I found to be a mediocre performance. She's much better in Pillow Talk, another romantic comedy and the only film of Day's I've liked so far.
As for other things I disliked, I found the plot and script to be extremely lacking. Even though the performances aren't great, the actors did the best they could with the material given. The film could have been better had the characters weren't so one-dimensional had more depth, like Jack Lemmon and Shirley MacLaine in The Apartment, for example.
Overall, That Touch of Mink isn't the worst film ever made, but it's certainly not one of the best. I'd recommend skipping this one.