Let me prove a weaker inequality, the proof of which allows further improvements. Note that $S(n,k)=h_{n-k}(1,2,\dots,k)$, where $$h_t(x_1,\dots,x_k)=\sum_{a_1+\dots+a_k=t,a_i\geqslant 0} x_1^{a_1}\dots x_k^{a_k}$$ is a complete symmetric polynomial. Thus $$\frac{S(n+1,k+1)}{S(n,k)}=\frac{h_{n-k}(1,2,\dots,k+1)}{h_{n-k}(1,2,\dots,k)}=\frac{\sum(p_1+1)\dots (p_{n-k}+1)}{\sum p_1\dots p_{n-k}},$$
where summation is taken over $0\leqslant p_1\leqslant p_2\leqslant \dots \leqslant p_{n-k}\leqslant k$. For any such $(n-k)$-tuple we have
$$(p_1+1)\dots (p_{n-k}+1)\geqslant \left(1+\frac1k\right)^{n-k}p_1\dots p_{n-k},$$
summing up we get $$S(n+1,k+1)\geqslant \left(1+\frac1k\right)^{n-k} S(n,k),$$
this is worse than you need, but the inequality was not sharp enough. Roughly speaking, you need to prove that "in average" the ratio $(1+1/p_1)\dots (1+1/p_{n-k})$ grows as $(1+2/k)^{n-k}$, that corresponds to "average" $p_i\sim k/2$.