We investigate the effect of constant-vorticity background shear on the properties of wavetrains in deep water. Using the methodology of Fokas (A Unified Approach to Boundary Value Problems, 2008, SIAM), we derive a higher-order nonlinear Schrödinger equation in the presence of shear and surface tension. We show that the presence of shear induces a strong coupling between the carrier wave and the mean-surface displacement. The effects of the background shear on the modulational instability of plane waves is also studied, where it is shown that shear can suppress instability, although not for all carrier wavelengths in the presence of surface tension. These results expand upon the findings of Thomas et al. (Phys. Fluids, vol. 24 (12), 2012, 127102). Using a modification of the generalized Lagrangian mean theory in Andrews & McIntyre (J. Fluid Mech., vol. 89, 1978, pp. 609–646) and approximate formulas for the velocity field in the fluid column, explicit, asymptotic approximations for the Lagrangian and Stokes drift velocities are obtained for plane-wave and Jacobi elliptic function solutions of the nonlinear Schrödinger equation. Numerical approximations to particle trajectories for these solutions are found and the Lagrangian and Stokes drift velocities corresponding to these numerical solutions corroborate the theoretical results. We show that background currents have significant effects on the mean transport properties of waves. In particular, certain combinations of background shear and carrier wave frequency lead to the disappearance of mean-surface mass transport. These results provide a possible explanation for the measurements reported in Smith (J. Phys. Oceanogr., vol. 36, 2006, pp. 1381–1402). Our results also provide further evidence of the viability of the modification of the Stokes drift velocity beyond the standard monochromatic approximation, such as recently proposed in Breivik et al. (J. Phys. Oceanogr., vol. 44, 2014, pp. 2433–2445) in order to obtain a closer match to a range of complex ocean wave spectra.