  We consider the interaction between passing to finite covers and ergodic
properties of the straight-line flow on finite area translation surfaces with
infinite topological type. Infinite type provides for a rich family of degree
$d$ covers for any integer $d>1$. We give examples which demonstrate that
passing to a finite cover can destroy ergodicity, but we also provide evidence
that this phenomenon is rare. We define a natural notion of a random degree $d$
cover and show that, in many cases, ergodicity and unique ergodicity are
preserved under passing to random covers. This work provides a new context for
exploring the relationship between recurrence of the Teichm\"uller flow and
ergodic properties of the straight-line flow.
