Mesoscale eddies contain the majority of oceanic kinetic energy and play a pivotal role in global heat transport, material exchange, and energy cascades. Submesoscale processes, serving as a dynamical bridge between mesoscale eddies and small-scale turbulence, are ubiquitous within eddies and are strongly modulated by wind stress. Down-front (Up-front) winds enhance (suppress) submesoscale instabilities by transporting dense (light) water over light (dense) water, thereby inputting positive (negative) Ekman buoyancy flux. Under a uniform wind field, both down-front and up-front winds can coexist within an anticyclonic eddy. Here, using submesoscale-resolving numerical simulations of an idealized anticyclone, we find that overall the wind forcing suppresses submesoscale activities within the eddy. Incorporating theoretical analysis and diagnostic analysis of the numerical output, it is demonstrated that although instabilities are more likely to develop in down-front regions, they are markedly suppressed in up-front regions. This suppression, under the influence of rotational advection of the vortex, reduces submesoscale activities over the entire anticyclone. Diagnostics of the global high-resolution MITgcm LLC4320 simulation also suggest that submesoscale activities within anticyclonic eddies are significantly in negative correlation with wind speed, confirming that wind stress exerts an overall suppressing effect on submesoscale processes in anticyclones.