  Let $\mathcal{I} = \{I_k\}_{k \in \mathbb{N}}$ be a graded family of monomial
ideal. We use the Newton-Okounkov body of $\mathcal{I}$ to: (a) give a
characterization for the Noetherian property of the Rees algebra of the family;
and (b) present a combinatorial interpretation for the analytic spread of the
family. We also apply these results to investigate the generation type and the
Veronese degree of the symbolic Rees algebra of a monomial ideal.
