In 1977, Gillette introduced Soft & Dri deodorant as their answer to Ban (then a property of Bristol-Myers, IIRC), and Procter & Gamble's Secret, a deodorant marketed to women. Gillette has subsequently sold the product, as it currently is made by Village Products.
In this 1979 spot, a then-unknown Rita Wilson (now Mrs. Tom Hanks) leads off a trio of vignettes to promote the product. Casey Kasem, who was everywhere at the time, and was a studio announcer for NBC, among his many jobs, is the narrator.
Nervous? I think they wanted you to buy the idea that it prevented wetness.