Eve was the first sinner because her action met all conditions required for sin. All of nature fell because of her...
If Adam had not sinned, Original Sin would not have been passed on to their offspring because of the sin of Eve. As the Apostles says, “Wherefore as by one man sin entered into this world, and by sin death; and so death passed upon all men, in whom all have sinned.”
Commentary found in the Douay Rheims Bible: "‘By one man’: Adam, from whom we all contracted original sin”
The Decrees on Original Sin from the Council of Trent
1. If anyone does not confess that the first man, Adam, when he transgressed the commandment of God in paradise, immediately lost the holiness and justice in which he had been constituted, and through the offense of that prevarication incurred the wrath and indignation of god, and thus death with which God had previously threatened him, and, together with death, captivity under his power who thenceforth had the empire of death, that is to say, the devil, and that the entire Adam through that offense of prevarication was changed in body and soul for the worse, let him be anathema.2. If anyone asserts that the transgression of Adam injured him alone and not his posterity, and that the holiness and justice which he received from God, which he lost, he lost for himself alone and not for us also; or that he, being defiled by the sin of disobedience, has transfused only death and the pains of the body into the whole human race, but not sin also, which is the death of the soul, let him be anathema, since he contradicts the Apostle who says:
‘By one man sin entered into the world and by sin death; and so death passed upon all men, in whom all have sinned’.3. If anyone asserts that this sin of Adam, which in its origin is one, and by propagation, not by imitation, is transfused into all, which is in each one as something that is his own, is taken away either by the forces of human nature or by a remedy other than the merit of the one mediator, our Lord Jesus Christ, who has reconciled us to God in his own blood, made unto us justice, sanctification and redemption; or if he denies that that merit of Jesus Christ is applied both to adults and to infants by the sacrament of baptism rightly administered in the form of the Church, let him be anathema; for there is no other name under heaven given to men, whereby we must be saved.Whence that declaration: ‘Behold the Lamb of God, behold him who taketh away the sins of the world;’ and that other: ‘As many of you as have been baptized, have put on Christ.’4. If anyone denies that infants, newly born from their mothers' wombs, are to be baptized, even though they be born of baptized parents, or says that they are indeed baptized for the remission of sins, but that they derive nothing of original sin from Adam which must be expiated by the laver of regeneration for the attainment of eternal life, whence it follows that in them the form of baptism for the remission of sins is to be understood not as true but as false, let him be anathema, for what the Apostle has said, by one man sin entered into the world, and by sin death, and so death passed upon all men, in whom all have sinned, is not to be understood otherwise than as the Catholic Church has everywhere and always understood it.”
It is a dogma of faith that Original Sin was passed on by the sin of Adam, not Eve. If Eve had sinned, but Adam did not, Original Sin would not have been passed on.