This is an attempt to rid of demand of continuity.
The following proof is essentially an adaptation of Banach's original proof in case of continuous function defined on segment $[a,b]$. See also https://math.stackexchange.com/a/144832/23566.

**Lemma** Let $f\colon \mathbb R^n \to \mathbb R^m$ be a measurable function and
an image $f(B)$ is measurable for any Borel measurable set $B$.
Then the Banach Indicatrix $N(y,f)$ is measurable.

**Proof.** We use a dyadic decomposition.
For each integer $k\geq 0$ consider collection of cubes $\{P^k_i\}$ of the form
$$
P^k_i = (a_1^i\cdot2^{-k},(a_1^i+1)\cdot2^{-k}]\times\cdots\times(a_n^i\cdot2^{-k},(a^i_{n+1})\cdot2^{-k}],
$$
where the $a^i_j$ are all integers.
The properties we need are following: cubes $P^k_i$ are disjoint; $\mathbb R^n = \bigcup\limits_{i=1}^{\infty}P^k_i$; $\operatorname{diam} P^k_i = \sqrt{n}2^{-k}\to 0 $ as $k\to\infty$.

For $y \in \mathbb R^m$ and $i\in \mathbb N$ let
$$
L_{i}^{(k)}(y) = \begin{cases} 1, & \text{if } y \in f(P_{i}^{(k)}), \\ 0, & \text{if } y \not\in f(P_{i}^{(k)}).
\end{cases}
$$
The functions $L_{i}^{(k)}(y)$ are non-negative and measurable because the set $f(P_{i}^{(k)})$ is measurable.
Therefore the sum
$$
N_k(y) =\sum\limits_{i=1}^{\infty}L_{i}^{(k)}(y)
$$
is also measurable.
Thus, the sequence $(N_k)_{k=1}^\infty$ of measurable functions is increasing and therefore
the pointwise limit
$$
N^*(y) = \lim_{k\to\infty} N_k(y)
$$
exists and is a measurable function of $y$.

Note that $N_k(y)$ simply counts on how many of the cubes $P_{i}^{(k)}$ the function $f$ attains the value $y$ at least once.
Thus $N(y,f) \geq N_k(y)$ for all $k$, so $N(y) \geq N^*(y)$.

Let us argue that $N^*(y) \geq N(y,f)$.
Let $q$ be an integer such that $N(y,f) \geq q$.
Then there exist $q$ different points $x_1,\dots,x_q$ such that $f(x_j) = y$.
If $k$ is so large that points $x_1,\dots,x_q$ are in separated cubes $\{P_{i_j}^{(k)}\}_1^q$
then $N_k(y) \geq q$.
This shows $N^*(y) \geq N(y,f)$ and thus $N^*(y) = N(y,f)$, establishing measurability.

**EDIT**

After a while I came to the following

**Theorem** Let $f:X\to Y$ be a $\mu_X$-measurable mapping, and $A\subset X$ be a Borel set.
Then $f$ can be redefined on a set of $\mu_X$-measure zero in such a way that
the Banach indicatrix $N(y,f,A)$ is a $\mu_Y$-measurable function.

Partly this was known though I've wrote a proof here.

Geometric Integration Theoryby S. Krantz and H. Parks. $\endgroup$