“I Should Care” was written by Axel Stordahl, Paul Weston and Sammy Cahn and published in 1944. It counts as a jazz standard and has been recorded by many artists.
The entry at the Jazz Standards website says this:
Bob Allen introduced “I Should Care” with the Tommy Dorsey Orchestra in MGM’s Thrill of a Romance in 1945. Soon after the release of the film Martha Tilton had a hit with the song, the first of four appearances it would make on the pop charts that year.
Frank Sinatra’s version was also released in 1945 as a 78 single with the B-side “When Your Lover Has Gone.”