Now, questions are swirling about foreign donations to the family's charitable foundation and whether that money influenced her work at the State Department, The Associated Press reported.
In the survey a majority of respondents believed that Clinton, the dominant figure in the 2016 presidential campaign, is strong and decisive, but only a minority said she is inspiring and likable.
Sixty-one percent said "honest" describes her only slightly well or not at all.
Nearly four in ten Democrats and more than six in ten independents agreed that "honest" was not the best word for her.
Even so, she is viewed more favorably than her potential Republican rivals, none of whom are as well-known as the former Secretary of State, senator and First Lady.
Republicans are trying to make her integrity central to the early 2016 campaign, painting her as a creature of Washington who flouts the rules to get ahead.
Clinton said last month that she used a personal account out of convenience. She deleted about 30,000 emails she has described as personal in nature and declined requests from congressional Republicans to turn over her server for an independent review.
A majority of respondents said they believe Hillary Clinton used a private address to shield her emails from transparency laws and they think she should turn her server over to a third party for further investigation, the AP report said.
The AP-GfK Poll of 1,077 adults was conducted online April 23-27 with a margin of error of 3.4 percent.