Closest I'm finding is a quote by Henry A. Wise Wood, referred to as a civic leader. Quoting Votes For Women: Woman Suffrage Movement by Marjorie Spruill Wheeler:
Women not only pointed out that women would cease to be womanly, but that male supporters of suffrage ("strong, masculine personalities") were inexplicably seeking to "demasculinize" government by diluting "with the qualities of the cow, the qualities of the bull upon which all the herd's safety must depend ..."
There's a similar account in Women and War by Jean Bethke Elshtain.
I can't access the book notes from either reference in Google books, but both references seem to offer a citation for the quote.
Edit 1: Digging a little deeper, it's not entirely clear to me who the quote's author was exactly. Wikipedia references a Henry Alexander Wise Wood and a Henry Wise Wood living at around the same time. The latter seems to have been more politically active. But neither of their entries references anything related to women suffrage debates.
Edit 2: Lars Borsten led me to the precise reference via his comment. The quote appears in the debate over the 19th amendment (p.12 in the doc). It's from Mr Henry A Wise Wood, then President of the Aero Club of America.
Aside: I can't resist throwing in this Gustave Le Bon quote in passing. He was one of the fathers of social psychology and an influential figure at the time. It perfectly illustrates the rampant sexism that prevailed in those days:
"There are a large number of women whose brains are closer in size to those of gorillas than to the most developed male brains. This inferiority is so obvious that no one can contest it for a moment; only its degree is worth discussion. All psychologists who have studied the intelligence of women... recognize today that they represent the most inferior forms of human evolution, and that they are closer to children and savages than to an adult, civilized man. They excel in fickleness, inconstancy, absence of thought and logic, and incapacity to reason. Without doubt there exist some distinguished women, very superior to the average man, but they are as exceptional as the birth of any monstrosity as for example, a gorilla with two heads; consequently, we may neglect them entirely." -- Gustave Le Bon, in a "Revue d'Anthropologie", 2nd Series, Vol. 2, 1879.