The following proposition appears in Peters and Steenbrink's book on Mixed Hodge Structures.
Proposition([Peters--Steenbrink, Proposition C.11]) If $f\colon X\to\Delta$ is proper and smooth over $\Delta^\ast:=\Delta-\{0\}$, then the inclusion $X_0\hookrightarrow X$ is a homotopy equivalence.
Does anyone know a reference for the proof of this result?
As mentioned in the comment, Clemens proved the case for normal crossing divisors. The general case will follow from this result, which I'll provide proof later.
Remark 1. Here $X_0$ being a normal crossing divisor means $X_0=\cup_im_iD_i$ and the $D_i$ are smooth proper varieties meeting transversely among each other and $m_i\ge 1$. The local equation of a point $x_0\in X_0$ in a neighborhood of $X$ is analytically equivalent to $t=x_1^{m_1}\cdots x_k^{m_k}$.
Note a normal crossing divisor is different from simple normal crossing divisor, in which the central divisor is reduced ($m_i=1$) and the family is called semi-stable. To get a semi-stable family from $X\to \Delta$, one need to take a base change with respect to a finite cover $p:\tilde{\Delta}\to \Delta$ where $\deg(p)$ is the l.c.m. of $m_i$. (People studying asymptotic Hodge theory prefer to work with a semi-stable family, that's why various literature, e.g., by David Morrison, on Clemens-Schmid sequence work with a semi-stable family, even though the existence of the limiting mixed Hodge structure only rely on normal crossing condition.)
Remark 2. A strong deformation retract $F:X\times I\to X$ means $F$ is a deformation retract ($F_0=F(\cdot,0)=Id_X$ and $F_1=F(\cdot,1)$ sends $X\to X_0$) and satisfies $F(x,t)=x$ whenever $x\in X_0$.
Now let's prove the general case based on Clemens' theorem.
Proof. By Hironaka, there is a log resolution $$\sigma: X\to Y,$$ where $\sigma$ is isomorphism over $Y\setminus Y_0^{sing}$ and $X_0=\sigma^{-1}(Y_0)$ is a normal crossing divisor.
We define $G:Y\times I\to Y$ by
$$ G(y,t)=\begin{cases}\sigma F(\sigma^{-1}(y),t),\: y\notin Y_0;\\ y,\:\:\:\:\:\:\:\:\:\:\:\: y\in Y_0. \end{cases} $$ I claim that $G$ is a strong deformation retract of $Y$ onto $Y_0$. In fact, it suffices to check the continuity. Since the points know where to flow outside $Y_0$, it suffices to check continuity on $Y_0$. Let's take a sequence of points $(y_n,t_n)$ on $Y\times I$ whose limit point is $(y_0,t_0)$ with $y_0\in Y_0$. Then $$\lim_{n\to \infty}G(y_n,t_n)=\sigma\lim_{n\to\infty}F(\sigma^{-1}(y_n),t_n).$$ Since $X$ is proper, $\sigma^{-1}(y_n)$ converges to a point $x_0\in \sigma^{-1}(y_0)\subseteq X_0$. By strongness of $F$, $F(\sigma^{-1}(y_n),t_n)\to (x_0,t_0)$, so $\lim_{n\to \infty}G(y_n,t_n)=(\sigma(x_0),t_0)=(y_0,t_0)$. Therefore $G$ is continuous on $Y_0\times I$. $\Box$
Final remark. One can think of the retraction as describing a flow that sends the vanishing cycles to the singularity on central fiber and send other points to smooth locus $Y_0^{sm}$. In the normal crossing case, the vanishing cycle is the boundary of a tubular neighborhood of $D_1\cap D_2$ (say two components) in $Y$, which is a circle bundle. The retraction is to flow each circle over $y\in D_1\cap D_2$ to the point $y$. To me, this is the topological picture behind the argument.
In addition to the references listed in this answer, one can refer to Clemens' original paper [Cle77] (especially Theorem 5.7) for the strong deformation retract theorem in normal crossing case. Besides, Nicolaescu's notes (e.g. Chapter 14) is also a good introduction to this subject.
[Cle77]: Clemens, C. H. Degeneration of Kähler manifolds. Duke Math. J. 44 (1977), no. 2, 215–290.