The primary goal of this research is to investigate the mechanical reinforcing efficiencies of carbon nanofibers in a thermoset polymer material (vinyl ester), considering the presence of the three-dimensional interphase region between nanofiber and matrix, as well as the waviness of the nanofibers. The elasticity-driven response (buckling) and energy absorption efficiency (crush performance) of the structures made of those composites are investigated. The structural/material optimization problem is solved for both buckling and energy absorption. Due to the nondeterministic nature of the influential parameters (fiber, matrix, and interphase geometric and material properties) on the overall properties of the composite, this study considers the probabilistically distributed random variables associated with the material constituents. The uncertainties associated with the material constituents are propagated to the overall properties of the composite material as well as the performance of composite structures made of such nanocomposites. Finally, the design optimization of a composite structure under uncertainty of material constituents is performed for both buckling and energy absorption as structural performance.