We analyze the r-->infinity asymptotic decay of the total correlation function h (r) for a fluid composed of particles interacting via a (point) Yukawa pair potential. Such a potential provides a simple model for dusty plasmas. The asymptotic decay is determined by the poles of the liquid structure factor in the complex plane. We use the hypernetted-chain closure to the Ornstein-Zernike equation to determine the line in the phase diagram, well removed from the freezing transition line, where crossover occurs in the ultimate decay of h (r) , from monotonic to damped oscillatory. We show that (i) crossover takes place via the same mechanism (coalescence of imaginary poles) as in the classical one-component plasma and in other models of Coulomb fluids and (ii) leading-order pole contributions provide an accurate description of h (r) at intermediate distances r as well as at long range.