Probably Spyros has in mind something like the following.
Suppose you have a semi-normalized basic sequence $(x_n)$ in $V$ with biorthogonal functionals $(f_n)_n$ in $V_0^\perp \subset V^*$. Take any normalized basic sequence $(y_n)_n$ in $V_0$. If $\epsilon_n \to 0$ sufficiently quickly with all $\epsilon_n >0$, then $(y_n + \epsilon_n x_n)_n$ is a basic sequence that is even equivalent to $(y_n)_n$ and $(\epsilon_n^{-1} f_n)_n$ are biorthogonal to $(y_n + \epsilon_n x_n)_n$. Since $(y_n + \epsilon_n x_n)_n$ is basic, $(f_n)_n$ separates the points in the closed linear span $W$ of $(y_n + \epsilon_n x_n)_n$, and hence $W \cap V_0 =\{0\}$. By construction, $W+V_0$ is not closed. (If it were closed, then the projection from $V$ to $V/V_0$ would take it isomorphically onto its image, but $\|y_n + \epsilon_n x_n\| \to 1$ and $\|\epsilon_n x_n\| \to 0$.)
To get such $(x_n)$ and $(f_n)$, pull back to a semi-normalized sequence $V$ any normalized basic sequence in $V/V_0$. That sequence in $V$ might not be basic, but you can pass to a subsequence of differences that is basic; see e.g. the early part of the chapter on basic sequences in the book of Albiac and Kalton.
