  We find an upper bound for the asymptotic dimension of a hyperbolic metric
space with a set of geodesics satisfying a certain boundedness condition
studied by Bowditch. The primary example is a collection of tight geodesics on
the curve graph of a compact orientable surface. We use this to conclude that a
curve graph has finite asymptotic dimension. It follows then that a curve graph
has property $A_1$. We also compute the asymptotic dimension of mapping class
groups of orientable surfaces with genus $\le 2$.
