Uncountably infinite as long as $d\ge2$.

We can solve the boundary value problem for countinuous functions on the unit sphere $S$.
So we get a linear injection from $C(S)$ to $H(B)$. Now the dimension of
$C(S)$ is uncountable. The unit interval $[0,1]$ embeds in $S$ and by Urysohn's
lemma we get a linear surjection from $C(S)$ to $C([0,1])$. To see that
$C[0,1]$ has uncountable dimension, note that the functions $f_t:x\mapsto e^{tx}$
are linearly independent for $t\in\mathbb{R}$.