Having been 7 and 8 years old during that time frame myself and obviously not very politically literate or attentive to current events at that age, I couldn't tell you one way or the other on that from first-hand experience. The first time I heard about that particular trial was in 2005 via a women's studies class that I was taking in college, and it was portrayed as I've described therein. However, I did watch family entertainment back in those days and still have my copies of such shows from that time frame and, knowing what I know now about that case, I can, in retrospect, see that certain episodes of popular family shows in circulation back then (like Dinosaurs, for example) were definitely referencing the Anita Hill show trial specifically in what for adult viewers would have been very unsubtle and disparaging ways. I have to suspect that there may have been some significant relationship between those portrayals and public opinion, as indicated by the subsequent election results.
In the women's movement, the Anita Hill trial is regarded as having marked the beginning of the "third wave" of feminist activism. (I don't consider myself personally to be a subscriber to so-called third wave philosophy, but my point is that it was an important moment in the history of the movement that brought about, among other things, a certain uptick in support for the movement.)