“Next to doing the right thing, the most important thing is to let people know you are doing the right thing.”

As quoted by John M. Kost http://www.mackinac.org/bio.aspx?ID=104 (25 July 1995) in S. 946, the Information Technology Management Reform Act of 1995: hearing before the Subcommittee on Oversight of Government Management and the District of Columbia of the Committee on Governmental Affairs (1996).
This appears to derive from a 1910 advertisement by writer Alfred Henry Lewis for a forthcoming series of biographical articles about Roosevelt: "All activity, Mr. Roosevelt has often shown that it is better to do the wrong thing than do nothing at all. In politics this last is peculiarly true. The best thing is to do the right thing; the next best is to do the wrong thing; the worst thing of all things is to stand perfectly still". (e.g. in La Follette's Magazine https://books.google.com/books?id=RV4CAAAAIAAJ&pg=PA183&dq=%22best+thing%22+%22right+thing%22+%22worst+thing%22+nothing&hl=en&sa=X&ved=0ahUKEwjNksu-nZrMAhVDy2MKHSl1Df8Q6AEIIzAB#v=onepage&q=%22the%20best%20thing%20is%20to%20do%20the%20right%20thing%22&f=false (28 May 1910)
Disputed

We The Living (1936)
Source: We The Living Part One Chapter 6

Source: Opportunity and Self-Transcendence (1977), p. 21

Source: Inner Engineering: A Yogi's Guide to Joy

Source: All the Light We Cannot See

Out of Control: The New Biology of Machines, Social Systems and the Economic World (1995), New Rules for the New Economy: 10 Radical Strategies for a Connected World (1999)

Misattributed
Variant: Efficiency is doing the thing right. Effectiveness is doing the right thing.
Source: The Essential Drucker

Peter Drucker, and Warren Bennis, as quoted in Seven Habits of Highly Effective People (1989) by Stephen R. Covey, p. 101
1980s

Peter Drucker, and Warren Bennis, as quoted by Covey, in The Seven Habits Of Highly Effective People (1989), this has sometimes become misattributed to him.
Misattributed