= Positive Bernstein coefficients for a strictly positive polynomial
If a polynomial $f$ is strictly positive on $[0,1]$, take $g_n=B_n^{-1}f$. By <inverse Bernstein approximation on a fixed-degree polynomial space>, eventually $g_n>0$ on the interval. Therefore
$$
f(x)=\sum_{k=0}^n\binom nk g_n(k/n)x^k(1-x)^{n-k}
$$
has strictly positive coefficients. The converse implication to nonnegativity follows immediately because each basis term is nonnegative. Strict positivity is essential for the general existence result: a nonzero polynomial vanishing at an interior point cannot have a nonnegative-coefficient representation in this basis, whose individual terms are positive throughout the open interval.
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