Is the space $B([a,b])$ separable?

The space in question is not separable, because you do not have any continuity assumption on the elements. For example, $C([a,b])$ is separable.

To see that your space is not separable, it suffices to construct an uncountable family $(f_i)_i$ in $B([a,b])$ such that $d(f_i, f_j) \geq 1$ for all $i \neq j$ (show this!!, let $(g_n)_n$ be dense in $B([a,b])$, take $\varepsilon = 1/2$ and note that for each $i$ there is some $n_i$ such that $d(f_i, g_{n_i}) < 1/2$. Why does this help you?).

To construct such a family, think for a few minutes or consider the spoiler below.

$$f_{x}\left(y\right)=\delta_{x,y}=\begin{cases}1, & x=y\\0, & x\neq y\end{cases} \text{ for each } x\in[a,b].$$

I leave it to you to check that actually $d(f_x, f_y) = 1$ holds for all $x \neq y$.