For complex manifolds, simultaneous contractions in a family have been studied by Riemenschneider in the paper "Deformations of Rational Singularities and their Resolutions". Theorem 1 in that article says that given a family of complex manifolds $\mathcal{X} \to T$ such that for some fiber $X_t$, there exists a contraction $\nu_t : X_t \to Z_t$ to a variety with at worst isolated singularities, then up to shrinking $T$, the map $\nu_t$ extends to a contraction $\nu : \mathcal{X} \to \mathcal{Z}$ over $T$.

If the base $T$ in your question is smooth, then up to shrinking $T$ the family version of Castelnuovo and Artin's contractibility criteria is a direct consequence of Theorem 1: Since $\mathcal{B}$ is isomorphic to $\mathcal{B}_t \times T$ over $T$ (up to shrinking $T$) and since $\nu(\mathcal{B}_t)$ is a point by assumption, each fiber of $\mathcal{B} \to T$ is contracted by $\nu$ to a point. As the contractions of $(-1)$ and $A$-$D$-$E$-curves are unique, the restriction of $\nu$ to each fiber $\mathcal{X}_s$ contracts $\mathcal{B}_s$ with the same type of contraction.

For complex manifolds, simultaneous contractions in a family has been studied by Riemenschneider in the paper "Deformations of Rational Singularities and their Resolutions". Theorem 1 in that article says that given a family of complex manifolds $\mathcal{X} \to T$ such that for some fiber $X_t$, there exists a contraction $\nu_t : X_t \to Z_t$ to a variety with at worst isolated singularities, then up to shrinking $T$, the map $\nu_t$ extends to a contraction $\nu : \mathcal{X} \to \mathcal{Z}$ over $T$.

Let me also mention the following well-known result concerning deformations of maps (see Theorem 2.1 in this article). If $\nu : X \to Z$ is a surjective map such that $\nu_* \mathcal{O}_X = \mathcal{O}_Z$ and $R^1\nu_*\mathcal{O}_X = 0$, then a small deformation $\mathcal{X} \to T$ of $X$ induces a deformation $\mathcal{X} \to \mathcal{Z}$ of $\nu$ over the same base.

From either of the two results above, if the base $T$ in your question is smooth, then up to shrinking $T$ the family version of Castelnuovo and Artin's contractibility criteria follows easily: Since $\mathcal{B}$ is isomorphic to $\mathcal{B}_t \times T$ over $T$ (up to shrinking $T$) and since $\nu(\mathcal{B}_t)$ is a point by assumption, each fiber of $\mathcal{B} \to T$ is contracted by $\nu$ to a point. As the contractions of $(-1)$ and $A$-$D$-$E$-curves are unique, the restriction of $\nu$ to each fiber $\mathcal{X}_s$ contracts $\mathcal{B}_s$ with the same type of contraction.