Does there exist an orthonormal basis of the Hilbert space $L^{2}(\mathbb{R}^{d})$, say $(e_{n})_{n=1}^{\infty}$, such that all elements $e_n\in L^{2}(\mathbb{R}^{d})\cap C^{0}(\mathbb{R}^{d})$ and such that for every $x\in\mathbb{R}^{d}$, the sequence $(e_{n}(x))_{n=1}^{\infty}\subset \mathbb{C}$ is in $\ell^{2}(\mathbb{N})$, i.e. such that the sum $$\sum_{n=1}^{\infty}|e_{n}(x)|^{2}$$ is finite?
I was thinking about the case $d=1$, and using the basis of $L^{2}(\mathbb{R})$ constructed from the Hermite polynomials, but I don't get enough decay to make the series converge.
This is only a partial answer but I believe that if the statement in question is incorrect---as @eyeballfrog suggested in their comment---then any proof would start with the following argument:
Assume there exists an orthonormal basis $(e_{n})_{n=1}^{\infty}$ of $L^{2}(\mathbb{R}^{d})$ such that $\sum_{n=1}^{\infty}|e_{n}(x)|^{2}<\infty$ for every $x\in\mathbb{R}^{d}$. Then the function \begin{align*} f:\mathbb R^d&\to\mathbb C\\ x&\mapsto\sum_{n=1}^\infty|e_n(x)|^2 \end{align*} is well-defined and, moreover, $f$ is the pointwise limit of the (obviously measurable) functions \begin{align*} f_\ell:\mathbb R^d&\to\mathbb C\\ x&\mapsto\sum_{n=1}^\ell|e_n(x)|^2\,. \end{align*} By Tonelli's theorem every $f_\ell$ is in $L^1(\mathbb R^n)$: \begin{align*} \int_{\mathbb R^d}|f_\ell(x)|\,{\mathrm d}\mu(x)&=\int_{\mathbb R^d}\sum_{n=1}^\ell|e_n(x)|^2\,{\mathrm d}\mu(x)\\ &=\sum_{n=1}^\ell\int_{\mathbb R^d}|e_n(x)|^2\,{\mathrm d}\mu(x)=\sum_{n=1}^\ell\|e_n\|_2^2=\sum_{n=1}^\ell 1=\ell<\infty\,, \end{align*} but $f$ is not ($\|f\|_1=\infty$). Therefore (the contraposition of) dominated convergence implies that $(f_\ell)_\ell$ cannot be dominated by any integrable function, i.e. for all $g\in L^1(\mathbb R^d,\mathbb R_+)$ and all $L\in\mathbb N$ there has to exist $x\in\mathbb R^d$ and $\ell>L$ such that $|f_\ell(x)|>g(x)$.
Intuitively, given $c>0$ I would choose something like $g_c:\mathbb R^d\to\mathbb C$, $x\mapsto c$ if $x\in B_c(0)$ and $x\mapsto c^3\|x\|^{-2}$ else so $g_c$ is integrable, positive, continuous, but diverges pointwise as $c\to\infty$. So far, however, I have not been able to turn this into a contradiction. Maybe someone else can complete this argument and solve the problem.