  Background: The density dependence of nuclear symmetry energy is crucial in
determining several properties of finite nuclei to the neutron stars with mass
$\sim$ 1.4 $M_\odot$. The values of neutron skin thickness, isovector giant
dipole resonances energies and various nuclear reaction cross-sections in
asymmetric nuclei have been utilized to determine the slope of symmetry energy
($L_0$) at the saturation density. Recent PREX-II and CREX measurements of
neutron skin thickness in $^{208}$Pb and $^{48}$Ca nuclei yield very different
values of $L_0$ which overlap marginally within 90$\%$ confidence interval.
  Purpose: Our objective is to demonstrate the role of symmetry energy on the
sub-barrier fusion cross-section and the astrophysical $S$-factor for
asymmetric nuclei.
  Method: The nucleus nucleus potentials are generated using the double folding
model (DFM) for three different nucleon-nucleon interactions. These DFM
potentials are used for the calculation of the sub-barrier fusion cross-section
and the astrophysical $S$-factor. The nucleon densities required for DFM
potentials are generated from different families of non-relativistic and
relativistic mean-field models which correspond to a wide range of neutron skin
thickness or $L_0$.
  Results: We have calculated the sub-barrier fusion cross-section for several
asymmetric nuclei involving O, Ca, Ni, and Sn isotopes. The results are
presented for the barrier parameters, cross-section, and astrophysical
$S$-factor for $^{54}$Ca+$^{54}$Ca and $^{124}$Sn+$^{124}$Sn as a function of
neutron skin thickness.
  Conclusions: The cross-section for the neutron-rich nuclei show a strong
dependence on the behavior of symmetry energy or the neutron skin thickness.
The increase in skin thickness lowers the height of the barrier as well as its
width which enhances the values of the $S$-factor by more than an order of
magnitude.
