The sinful nature was not necessarily inherited. The idea that Adam passed sinfulness to his descendants like an infection is one of the traditional metaphors in the doctrine of original sin (Tertullian, Anselm, Luther, Augsburg Confession). The problem with this conception is that it doesn’t take into account mankind’s voluntary disposition and responsibility (guilt) for sinning, which make it qualitatively different from sickness.
A second metaphor for the transmission of sin from Adam to his posterity is legal, drawn from Paul’s discussion in Romans 5. This is the “federal” view of Reformed theology, which pictures Adam as the representative of us all, and thus his guilt is imputed to us all. The strengths of this metaphor are that it does not rest on physical transmission of sin from Adam to his posterity, and it is more consistent with the language of Scripture. The problem is that it violates many people’s (including Christians) sense of justice: How could God declare us “guilty” on the basis of a choice we had nothing to do with?
The third common metaphor is one of privation, or lack. Catholic theology since Anselm has preferred this approach. The fall entailed the loss of original righteousness and other divine gifts. If Adam lost the gifts, obviously he could not pass them down to his descendants.
The French evangelical theologian Henri Blocher has an excellent book Original Sin: Illuminating the Riddle, if you are interested in the topic.