**Case 1:** Let the liquid be volatile say water. The liquids are mostly volatile. The fan blows air over the liquid surface. This makes the liquid to evaporate & taken away in form of vapors by blowing air. In this process, the liquid molecules on free surface of liquid takes enthalpy of vaporization partly from neighing molecules and partly from blowing air to change in vapor. This results in heat extraction & cooling of volatile liquid and also possibly partial cooling of blowing air depending on temperature of liquid. The minimum temperature that a volatile liquid can achieve is the Thermodynamic Wet Bulb Temperature (TWBT) of ambient air.
This is one of reasons why water (i.e. abundant volatile liquid with high enthalpy of vaporization) is used in water cooling systems. 

The condition of a volatile liquid to cooled by air blowing over its surface is that the temperature of volatile liquid should be greater than dew point temperature (DBT) (i.e. temperature at which water vapor in air starts condensing) & less than wet bulb temperature (WBT) (i.e. temperature at which air is saturated with vapor) of blowing air.

**Case2:** Let liquid be non volatile (i.e. can't evaporate) say Glycerin. The air blown by fan extracts heat from liquid surface by (forced) convection process if the temperature of (ambient) blowing air is less than liquid temperature & it can cool non-volatile liquid (Glycerin) otherwise not.