$\newcommand\ep\epsilon$In view of the [Berry--Esseen inequality][1] and the inequality $\sum_{i=1}^n E|X_i|^3\le s_n^2$, 
$$P(1+|S_n|\le\epsilon s_n)\le
1(1\le\ep s_n)\Big(P(|Z|\le\ep)+\frac1{s_n}\Big)
\le 1(1\le\ep s_n)\Big(\ep+\frac1{s_n}\Big)\le2\ep, $$
where $Z$ is a standard normal random variable. 

So, $P(1+|S_n|\le\epsilon s_n)\to0$ uniformly in $n$ as $\ep\downarrow0$. Thus, $P(1+|S_n|>\epsilon s_n)\to1$ uniformly in $n$ as $\ep\downarrow0$. 


  [1]: http://Berry--Esseen%20inequality