In this paper, we consider the Cauchy problem of the three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities. When the shear and bulk viscosity coefficients are both given as a constant multiple of the mass density's power ( ${\rho }^{\delta }$ with $\delta >1$), we show that the $L^{\mathrm{\infty }}$ norms of $\mathrm{\nabla }u$, $\mathrm{\nabla }{\rho }^{\frac{\gamma -1}{2}}$ and $\mathrm{\nabla }{\rho }^{\frac{\delta -1}{2}}$ control the possible breakdown of classical solutions with far-field vacuum; this criterion is analogous to Serrin's blowup criterion for the compressible Navier–Stokes equations.