Let $v$ be a vector obtained by taking a sum of $k$ vectors the of the form $(0,0,\ldots,0, -n, *,*,\ldots,*)$, where $"*"$ stands for either $0$ or $1$, and the position of the $-n$ entry can vary for each summand of $v$. We also require the sum of all entries of $v$ to equal to zero.
Is it possible that $$\frac{\|v\|_1}{k\cdot n}$$ is arbitrary small?
Let $n=1$, fix $k$, and consider the $k$ vectors \begin{eqnarray} v_1 &=& (-1,1,0,0,0,0,\ldots) \\ v_2 &=& (0,-1,1,0,0,0,\ldots) \\ v_3 &=& (0,0,-1,1,0,0,\ldots) \\ v_4 &=& (0,0,0,-1,1,0,\ldots) \\ \vdots & \vdots & \vdots \end{eqnarray} Here, each $v_i\in \mathbb{R}^d$ where $d>k$. Let $v=v_1+\cdots+v_k$. Then $v=(-1,0,0,\ldots,0,1,0,\dots)$, $||v||_1 = 2$, and $\frac{||v||_1}{nk}\to 0$ as $k\to\infty$. This is easily generalized to $n>1$ if you want.