I once heard a young woman tell me, "women shouldn't take the initiative in relationships because it was Eve who took the initiative in Eden and caused the fall." I think her point was, not that women shouldn't take the initiative because they were responsble for the fall, but that the fall proved that women were never meant to take the initiative. From there it seems a short step to this kind of thinking,
It is fair to say that there are a number of women in Genesis who take the initiative with negative results and in ways that disobey God. Apart from Eve, the examples of Sarai telling Abram to impregnate Hagar (Gen 16) and Potiphar's wife attempting to seduce Joseph (Gen 39) come to mind. However, in none of those examples is there any indication that the problem with their actions is that they are women. In each instance those actions are simply expressions of sin regardless of gender. Adam, Abram, and Potiphar, hardly serve as shining examples of ethical initiative taking in those narratives either.
On the other hand, in two significant narrative sections women take the initiative in highly unorthodox ways, and yet, within the world view of Genesis, are richly rewarded. The first story is that of Lot's daughters, to our modern minds a tale of depravity and incest, but in Genesis actually the story of how Lot's daughters take the initiative when their father is too scared and useless to find them husbands and ensure the survival of the family by nay means necessary. Their reward is that their offspring become two great nations, the Moabites and the Ammonites, the women are vindicated by history (Gen 19:30-38). The second story is of Judah and Tamar, where in a scarcely less sordid episode Judah's daughter in law has to disguise herself as a prostitute in order to seduce him and preserve the family line. She is rewarded by success and the recognition that her actions were righteous (Gen 38, not to mention Matt 1:3). None of this is to suggest that women's place is only in preserving the family line, i.e. breeding, but in Genesis that is pretty much all anyone, male or female, cares about. The point is that women take the initiative throughout Genesis, sometimes it results in bad and sometimes in good, just like it does with the men.
If anything, Gen 3 actually shows the Eve taking some responsibilty and admitting being deceived (3:13), whereas Adam just tries to blame both Eve and God for putting Eve in the garden with him (3:12).