“The public cannot be too curious concerning the characters of public men” is often credited to Samuel Adams (1722-1803), who wrote this in a letter to James Warren, dated November 4, 1775. Quotation marks were used by Adams.
 
The statement was first used by James Burgh (1714-1775) in his Political Disquisitions (published in three volumes in 1774-1775).
         
 
Wikipedia: James Burgh
James Burgh (1714–1775) was a British Whig politician whose book Political Disquisitions set out an early case for free speech and universal suffrage: In it, he writes, “All lawful authority, legislative, and executive, originates from the people.” He has ben judged “one of England’s foremost propagandists for radical reform.”

Burgh also ran a dissenting academy and wrote on subjects such as educational reform. One of his first books was Thoughts on Education (1747). His widow acted as fairy godmother to early feminist Mary Wollstonecraft, then a young and unpublished schoolmistress, who titled her first book Thoughts on the Education of Daughters (1787). Burgh’s title in turn alludes to John Locke’s 1693 work, Some Thoughts Concerning Education.
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Political Disquisitions (1774)
Of the three volumes of Political Disquisitions, the third is the most widely referenced. The book was inspired by the radical reform movement of the time, and includes many of Burgh’s feelings on social, religious, political and educational reforms. Burgh also includes many other authors in the book, with the strongest influence being that of John Locke.
 
Wikipedia: Samuel Adams
Samuel Adams (September 27 [O.S. September 16] 1722 – October 2, 1803) was a statesman, political philosopher, and one of the Founding Fathers of the United States. As a politician in colonial Massachusetts, Adams was a leader of the movement that became the American Revolution, and was one of the architects of the principles of American republicanism that shaped the political culture of the United States. He was a second cousin to John Adams.
 
Google Books
Political disquisitions: or, An enquiry into public errors, defects, and abuses; illustrated by ans established upon facts and remarks extracted from a variety of authors, ancient and modern
By James Burgh
London: Pr. for E. and C. Dilly
1774
Pg. 11:
The public cannot be too curious concerning the characters of public men; so common is it for them to change upon preferment, according to the old adage, honores mutant mores.
 
Google Books
The writings of Samuel Adams
By Samuel Adams
Edited by Harry Alonzo Cushing
New York, NY: G.P. Putnam’s Sons
1904-08
Pg. 237:
TO JAMES WARREN.
[MS., Samuel Adams Papers, Lenox Library.]
PHILAD Novr 4 1775
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“Nothing is more essential to the Establishment of manners in a State than that all Persons employd in Places of Power and Trust be Men of unexceptionable Characters. The Publick cannot be too curious concerning the Characters of publick Men.”
 
13 November 1775, Pennsylvania Packet (Philadelphia, PA), pg. 3:
PHILADELPHIA, November 13.
The several printers in the united colonies are desired to publish in their news-papers, the following extracts from BURGH’s political disquisitions, for the benefit of those who have not these useful books.
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Nothing is more essentially necessary to the establishment of manners in a state, than that all persons employed in stations of power and trust be men of exemplary characters: The public cannot be too curious concerning the characters of public men.
   
Glenn Beck (glennbeck) on Twitter
“The public cannot be too curious concerning the characters of public men.” Samuel Adams, learn more this Founding Father’s Friday
(April 28, 2010—ed.)