...what do you think it would be?

If you mean Harriet the real person, she was an actress/singer and dnacer. If you mean her TV character, 1950s women were mostly homemakers, but she probably would have been a secretary or teacher. They put June in as a temporary secretary, so I suppose that ould have been the most acceptable job for her too.

In the radio show, they made mention that she was once the singer in Ozzie's band, and since the TV show was a carry-over of the radio show, I work under the assumption that she's a former singer.

Though being married to Ozzie and having to solve his, Rick's and Dave's problems is a full-time job, let alone having the Tigers drop by asking for cookies, or Clara Randolph asking Ozzie to come over and impersonate her husband or....

If you are interested in Harriet Nelson's actual career prior to the TV show, the book "Ricky Nelson: Idol for A Generation" is an excellent source. It's written by Joel Selvin and I found lots of copies for sale at abebooks.com. abebooks.com also has plenty of Ozzie's biography for sale at very low prices.

...as an actress and singer (primarily with Ozzie's band, until they started their own radio show in 1944). Once in a while, Harriet might have referred to her previous career in any given episode, but, like Ozzie, that was rarely noted.