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 $t\in\mathbb{R}$.
Added A less clumsy way of proving that $C(S)$ is uncountably dimensional than the hasty argument above is to consider $F_t:(x_1,\ldots,x_n)\mapsto e^{t x_1}$.