Why don't minerals evaporate from seawater along with the water?
In order to get a measure of how much a compound is prone to evaporation one can check boiling points. Boiling point of water is 100 C in atmospheric pressure, for molecular NaCl, sodium sulfate and Magnesium chloride (main minerals in seawater) boiling point is above 1400 C. For ionic salts dissolved in water there is a different story since they are ionized. The ions are surrounded by water molecules in liquid phase, while if vaporized they would not have a chance of having water molecules closeby so they prefer to stay in the liquid phase. All of that being said, one can envision a certain temperature, pressure and concentration (extremely high for three) at which ions would somehow get out to the vapor phase. Since sea level temperature is quite low, mineral or salt evaporation is practically zero. 

Side note on why would water evaporate at 4 C instead of 100 C: 
Although it take 100 C of kinetic energy (molecule's velocity) to have "all" water molecules to escape from the intermolecular attractive forces, still "some" of them end up having the energy to escape at 4 C (sea temperature average). It is because kinetic energy has distribution over all the molecules, and you will always find some that have the necessary energy. At the same time "some" of the water molecules in the air end up not having enough kinetic energy to remain in the air, and so they end up being attracted back to the liquid. These two numbers (The molecules of water in air that condense back and the water molecules in the ocean that evaporate) are equal so we have thermodynamic equilibrium. The net number of molecules that escape the sea depends on kinetic energy distribution which is exemplified in temperature.