Answered by David Mitra in the comments:
From Diestel's Sequences and Series in Banach Spaces, page 6: "R. C. James (1964) proved that a Banach space $X$ is reflexive if and only if each $x^*\in X^*$ achieves its norm on $B_X$. Using this, one can establish the following: For a Banach space $X$ to have the property that given a proper closed linear subspace $Y$ of $X$ there exists an $x$ of norm-one such that $d(x,Y)\ge 1$ it is necessary and sufficient that $X$ be reflexive."
A specific counterexample to your claim is: Let $X$ be the subspace of $C[0,1]$ consisting of the members of $C[0,1]$ that vanish at $0$. Let $Y$ be the subspace of $X$ consisting of those functions $f$ for which $\int_0^1 f(x)\,dx=0$. Then there is no $x\in S_X$ with $d(x,Y)\ge 1$ (and thus no $x\in B_X$ with $d(x,Y)\ge 1$).
For completeness: for any $x\in B_X$ we have $$\left|\int_0^1 x(t)\right|\,dt<1 \tag1$$ and the left side of (1) is nothing but $d(x,Y)$.