Danaher makes automotive and transportation products, instruments and precison components. Yesterday, it reported gains in earnings for 1989 and the fourth quarter. But in announcing the results, Steven Rales said the company ''remains cautious about the current year.'' Mr. Sherman was the No. 2 executive at Black & Decker behind Nolan D. Archibald, the president and chief executive who earlier this month added the title of chairman. Mr. Sherman joined Black & Decker, which is based in Towson, Md., in 1985, when the company's power tools were losing market share to both Japan and West Germany. He is given credit with leading a turnaround that brought back market share worldwide. Mr. Sherman has an engineering degree from Long Island University and an M.B.A from the University of Lousiville in Kentucky. He began his management career at the General Electric Company and spent 13 years there, rising to product general manager in the housewares and audio division.