The set of embedded minimal surfaces $\Sigma$ with $\textrm{genus}(\Sigma) \leq g_0$ and $\textrm{area}(\Sigma)\leq A_0$ is compact for a generic metric (this follows from White's version of the Choi--Schoen compactness theorem https://mathscinet.ams.org/mathscinet-getitem?mr=880951, see also Appendix A: https://mathscinet.ams.org/mathscinet-getitem?mr=1778099). 

*However*, without the area bounds, this fails: https://mathscinet.ams.org/mathscinet-getitem?mr=1728019 (generic metrics without compactness of stable spheres), https://mathscinet.ams.org/mathscinet-getitem?mr=2026836 (same for higher genus), https://mathscinet.ams.org/mathscinet-getitem?mr=2103999 (positive scalar curvature). In the first two papers, the surfaces constructed are also stable.