The following stronger inequality holds for all $k=0,\dots,n$: 
$$
(n+1)\binom{n}{k}\int_{0}^{\frac{k}{n+1}}t^k(1-t)^{n-k}\,dt \le 1/2.
$$
Indeed, the latter inequality means precisely that the median $m$ of the Beta distribution with parameters $a:=k+1\ge1$ and $b:=n-k+1\ge1$ is no less than $\frac{k}{n+1}=\frac{a-1}{a+b-1}$. 
By the well-known "Mode, Median, and Mean Inequalities" for the Beta distribution (see e.g. page 2 in http://arxiv.org/abs/1111.0433v1), 
\begin{equation}
	m\ge\frac{a-1}{a+b-2}\bigwedge\frac{a}{a+b} \ge\frac{a-1}{a+b-1}=\frac{k}{n+1},  
\end{equation}
as desired.