Let $X$ be an irreducible complex affine variety and $W \subset X$ a variety of dimension strictly less than $X$. It follows from Mumford, Algebraic Geometry I, Corollary 4.16 that $X \backslash W$ is connected in the Euclidean topology. Let $g: X' \rightarrow X$ be a desingularization of $X$.
Is it true that $g^{-1}(X \backslash W)$ is connected?
Yes, this is true. As $g$ is birational, $X'$ is irreducible; as $g^{-1}(X\setminus W)=X'\setminus g^{-1}(W)$, if we can see that if the closed subset $g^{-1}(W)$ of $X$ is a proper subset of $X'$ then we can apply the same result of Mumford you record in your question to see that $X'\setminus g^{-1}(W)$ is connected. But this follows from birationality of $g$ and irreducibility of $X$: $g$ is an isomorphism on some open subset of $X\setminus W$, and in particular there is some point of $X'$ not in $g^{-1}(W)$.