I claim that the answer is $ \lceil \frac{n^2}{2} \rceil$. Indeed, consider a chess coloring with $ \lceil \frac{n^2}{2} \rceil$ black squares and $ \lfloor \frac{n^2}{2} \rfloor$ white squares. We put distinct positive reals in the black squares in the interval $(1,2)$, and distinct positive reals in the white squares in the interval $(0,1)$. We notice that it is impossible for water to flow from a white square to a black one; therefore, we need at least one water source for each of the $ \lceil \frac{n^2}{2} \rceil$ black squares in this case. Now, to see that this is enough for all possible configurations, partition the $n \times n$ board into $ \lceil \frac{n^2}{2} \rceil$ regions of size $2 \times 1$ (with the exception of possibly one $1 \times 1$ rectangle in the case where $n$ is odd). For each of these regions, we put a water source in the maximum height in them. Just putting one in each region is enough to cover the whole garden, since the water will always flow from the maximum square to the minimum one in each of the regions.