From a chemical engineering perspective (where we do a lot with steam), the answer is dependent on if you're in a closed system or not. If I put some water in a closed container, it would evaporate only enough such that the gaseous water would reach the vapor pressure at that specific temperature. All temperatures above $\pu{0 K}$ have a non-zero vapor pressure, so you could say that water evaporates at all temperatures, at least for a while, and then it will be in equilibrium with the liquid (or solid) state. However, if you are not in a closed system (say, outside with a cup of water), then the water will continue to evaporate to reach the vapor pressure, but the gas will continue to escape, so the water will never stop evaporating.
The opposite, and the answer to your question, is also true. If you have circumstance when the surrounding pressure due to water vapor is higher than the vapor pressure of water at that temperature (like a rain cloud cooling down), then the water will start to precipitate. Of course, like all equilibrium, even if the current net phase shift is from gas towards liquid water, some of the liquid will still be evaporating - it's just that more gas will be condensing.