If the market for water is free, a drought will cause the price of water to go up. It is possible that drought will not cause the price of water to go up because governments will not allow this to happen. However, if the market is left to set the price of water on its own, a drought will lead to an increase in the price of water.
The price of a good in a free market is set by the forces of supply and demand. In order for the price of a good (in this case water) to rise, demand for that good must rise and/or the supply of that good must fall. There is no real reason why a drought should affect the demand for water. People will still need and want the same amount of water for bathing, cooking, watering their lawns, and other such things. The drought does not change this in the short term. Drought does, however, greatly decrease the supply of water. This is the very definition of a drought. When there is a drought, less water is available. When less water is available, the supply of water drops. When the supply of a good drops, the price of that good will (all other things being equal), rise.
Thus, droughts cause a decrease in the supply of water and, thereby, an increase in the price of water in a free market.