We investigate linear exciton transverse transports in monolayer two-dimensional semiconductors within the semiclassical transport framework. By deriving the exciton Berry curvature in momentum space for a general inhomogeneous two-dimensional system, we show that it is determined by both the Berry curvatures and effective masses of the constituent electron and hole. As representative examples, we analyze the exciton Hall and Nernst responses in monolayer transition-metal dichalcogenides (TMDs) and black phosphorus (BP). We find that, in these systems, a linear Berry-curvature-driven exciton Hall or Nernst current is strictly forbidden by symmetry, consistent with previous experimental observations of exciton Hall transport in monolayer MoSe${}_2$. In contrast, despite its vanishing exciton Berry curvature, monolayer BP supports a sizable linear transverse exciton current arising from its strong effective-mass anisotropy and the non-equilibrium part of the exciton distribution function in the Boltzmann equation. Our results show that crystalline anisotropy can generate a significant linear exciton transverse response in monolayer semiconductors without relying on Berry-curvature effects. This mechanism provides a distinct route toward controlling exciton transport and may offer useful guidance for future optoelectronic applications and experimental studies of excitonic transverse responses.