This is basically a comment on Dario's answer. I'm going to compare the Dirichlet problem on $B$ with the one on $B_0\equiv B\setminus\{ 0\}$ (though I'm not going to justify formally that this is what the limit in the OP will give), and Dario's argument will show that these are identical. In particular, all eigenvalues agree.

Recall that we may obtain the Dirichlet Laplacian on an open set $\Omega\subset\mathbb R^n$ from its quadratic form
$$
Q(f) = \int_{\Omega} |\nabla f|^2\, dx , \quad\quad f\in H_0^1(\Omega) ;
$$
in other words, $H_0^1$ is the domain of $(-\Delta_D)^{1/2}$, and $\Delta_D$ denotes the self-adjoint operator on $L^2(\Omega)$ we are interested in here (the Dirichlet Laplacian). See for example here for background.

Now in the situation we are interested in and for $n\ge 3$, we have that
$$
H_0^1(B)=H_0^1(B_0) \quad\quad\quad\quad (1)
$$
(as subspaces of $L^2(B)$, say).

(1) is established by the argument given by Dario. Since $H_0^1(\Omega)$ may be obtained as the closure of $C_0^{\infty}(\Omega)$ under the Sobolev norm $(Q(f)+\|f\|^2)^{1/2}$, it suffices to show that any $f\in C_0^{\infty}(B)$ can be approximated by smooth functions whose support is separated from $0$. This we do as in Dario's answer by taking $f_n(x)=\chi_n(|x|) f(x)$, with $0\le \chi_n\le 1$, $\chi_n=0$ near $0$, $\chi_n(r)=1$ for $r\ge 1/n$ and $\chi'_n\lesssim n$.