As is well known, light arising from Raman scattering on molecules shows nonclassical mutual correlations between the Stokes and anti-Stokes components. However, the practical use of nonclassical Raman light is limited, due to its low intensity. The Raman signal can be enhanced in surface-enhanced Raman scattering (SERS) experiments by placing Raman active molecules near a plasmonic nanoantenna, due to the Purcell effect. Since the main contribution to emission in SERS experiments is re-emitted light from the plasmonic nanoantenna with no nonclassical correlations, the question arises as to whether it is possible to simultaneously amplify the Raman signal and preserve the nonclassical correlations. In this work, we give a positive answer to this question. We show that plasmonic nanoantennas can be used to enhance the nonclassical light intensity. We demonstrate that the main factor causing deterioration of the cross correlations of SERS light is the fluorescent background of the plasmonic nanoantenna. We analyze the way in which the geometry of a plasmonic nanoantenna affects the nonclassical cross correlations of the Raman light, and show that there is an optimal size of plasmonic nanoantenna that both increases the Raman signal and preserves the nonclassical correlations of the light at the level of a single molecule. The obtained results pave the way for the creation of sources with controlled intensity and a degree of nonclassicality.