Water evaporates, at all temperatures. This is because evaporation is a surface phenomenon. On surface, some of the water molecules have enough kinetic energy to become free and transform to the gas phase. This can be explained well by the Maxwell Boltzmann distribution curve.
The distribution of molecules at different temperatures shows that some of the molecules have kinetic energy so high that they can escape from the surface. The number of molecules with such high energy continues to increase with an increase in temperature. It means that at a particular temperature the number of molecules present in the area of curve for activation energy determines the evaporation rate.
This curve also shows that evaporation continues at all temperatures because there will be at least some molecules on the surface that have enough kinetic energy to evaporate. In other words, there will be at least some portion of the Boltzmann curve in the activation energy curve portion.