The Dead Sea is below sea level, this means that it does not drain so that although the water evaporates, the salt will stay, this means that the dead sea is a saturated solution.
You can see this because salt crystals appear around the shore line.
The oceans become salty for the same reason - soluble salts are washed in to the oceans by rivers. Water from the oceans evaporates - leaving behind the less volatile salts.
The oceans are not saturated solutions because they are so vast.
The Dead Sea is relatively very small and land locked.
Evaporation causes large deposits of solid salt to form around the shore or the Dead Sea