Let $\{x_n,n\geq1\}$ be a sequence of positive numbers and $\{\xi_n,n\geq1\}$ be a sequence of nonnegative negatively orthant dependent (NOD) random variables satisfying certain distribution conditions. An exponential inequality for the minimum $\min_{1\leq i\leq n}x_i\xi_i$ is given. In addition, the moment inequalities of the minimum $(\Bbb Ek-\min_{1\leq i\leq n}|x_i\xi_i|^p)^{1/p}$ for nonnegative negatively orthant dependent random variables are established, where $p>0$ and $k=1,2,\cdots,n$. Our results generalize the corresponding ones for independent random variables to the case of negatively orthant dependent random variables.