Probably this was downvoted because it is follows from standard material in most algebraic geometry textbooks. It might be more appropriate for Math Stack Exchange.

At any rate, with the question as formulated, this locus is not an open subset. For instance, consider a copy of $\mathbb{C}$ with coordinate $t$ inside your parameter space, where $f$ and $g$ are $f=x_1(1-tx_3)$ and $g=x_2(1-tx_3)$. For $t\neq 0$, then $V_{f,g}$ contains the codimension one hypersurface $\{(x_1,x_2,x_3) : x_3= 1/t\}$. But for $t=0$, $V_{f,g}$ has codimension $2$. Thus the intersection of this copy of $\mathbb{C}$ with your locus is the Zariski closed subset $\{t : t=0\}$, rather than a Zariski open subset. This type of example is one reason that algebraic geometers frequently work with proper / projective varieties rather than affine varieties.