We numerically investigate linear and nonlinear optical responses in metasurfaces consisting of Au double-gap split ring resonators (DSRRs). Symmetry-protected dual bound states in the continuum (BICs) in such plasmonic metasurfaces are observed at the near-infrared optical regime. Efficient second harmonic generation (SHG) is obtained at the quasi-BIC models due to the symmetry broken. The optimized SHG responses are obtained at the critical couplings between radiation and nonradiation processes at the linearly x- and y-polarized light, respectively. High conversion efficiency of SHG of a value 10-6 is arrived at the fundamental intensity of 10 GW/cm2 at the quasi-BIC wavelength under the y-polarized illumination. Large extrinsic and tunable chirality of linear and nonlinear optical responses empowered by quasi-BICs is acquired in asymmetry metasurfaces at oblique circularly polarized incidence. The results indicate that the plasmonic metasurfaces of symmetry-protected BICs at the near-infrared optical regime have great potential applications in the on-chip efficient frequency conversion, and the linear and nonlinear chiral manipulation.