Bounds in the univariate case, see e.g. here, were established by V.A. Markov in 1892.

S.N. Bernstein has given an extension of the result to the multivariate case in

S.N. Bernstein, On certain elementary extremal properties of
polynomials in several variables, Doklady Akad. Nauk SSSR (N.S.) 59
(1948), 833-836.

Unfortunately I do not have access to that paper, but according to MR0023953, the following result is proved. Let
$$P(x_{1},\ldots,x_{n})=\sum_{1\leq\alpha_{h}\leq d_{h}} A_{\alpha_{1},\ldots,\alpha_{n}}x_{1}^{\alpha_{1}}\ldots
x_{n}^{\alpha_{n}},$$
such that $|P|\leq1$ on the cube $|x_{h}|\leq1$, $1\leq h\leq n$. Then
$$|A_{\alpha_{1},\ldots,\alpha_{n}}|\leq\frac{\prod_{h=1}^{n}|B_{\alpha_{h}}^{(d_{h})}|}{\alpha_{1}!\ldots\alpha_{n}!},$$
where $B_{\alpha_{h}}^{(d_{h})}$ is the coefficient of $x^{\alpha_{h}}/\alpha_{h}!$ in the expansion of the Chebyshev polynomial
$$T_{m}(x)=\cos(m\cos^{-1}(x))=\sum_{i=0}^{m}B_{i}^{(m)}x^{i}/i!,$$
and $m=d_{h}$ if $d_{h}-\alpha_{h}$ is even and $m=d_{h}-1$ if $d_{h}-\alpha_{h}$ is odd.

For the particular case of homogeneous polynomials
$$P_{d}(x_{1},\ldots,x_{n})=\sum_{|\alpha|= d}c_{\alpha}x^{\alpha},$$
of total degree $d$, where $\alpha$ are multi-indices, such that
$|P_{d}|\leq1$ on the ball $x_{1}^{2}+\cdots +x_{n}^{2}\leq1$, upper bounds on the coefficients are given in Theorem 2 of

O.D. Kellogg, On bounded polynomials in several variables, Math. Z.
27 (1928), no. 1, 5-64.

namely, the coefficient of the monomial $x_{1}^{k_{1}}\ldots x_{n}^{k_{n}}$ cannot exceed in absolute value
$$\frac{d!}{k_{1}!\ldots k_{n}!}.$$

For general (nonhomogeneous) polynomials of total degree $d$, there are also precise bounds
on the coefficients $c_{\alpha}$ with $|\alpha|=d$ or $d-1$ (for the case of the ball or the cube).

In that connection, two interesting papers are

M.I. Ganzburg, A Markov-type inequality for multivariate polynomials
on a convex body. J. Comput. Anal. Appl. 4 (2002), no. 3, 265-268

H-J. Rack, On V. A. Markov's and G. Szegő's inequality for the
coefficients of polynomials in one and several variables. East J.
Approx. 14 (2008), no. 3, 319-352,

and the bibliography therein.