The Wikipedia article that you are looking at is, not about the Komlós–Major–Tusnády (KMT) approximation for sums of iid r.v.'s, but about the KMT approximation for the empirical process. However, the second paper (KMT 1976) cited in the Wikipedia article contains the following result. 

Let $S_n:=\sum_1^n X_i$, where the $X_i$'s are iid r.v.s such that $EX_1=0$, $EX_1^2=1$, and $Ee^{t|X_1|}<\infty$ for some $t>0$. Then one can construct all the $X_i$'s and a standard Wiener process $W(\cdot)$ on the same probability space so that for some positive real $C,K,\lambda$ and all natural $n$ and all real $x>0$ one has 
$$
P(\sup_{1\le k\le n}|S_k-W(k)|>x+C\ln n)\le Ke^{-\lambda x}  
$$
$\Big($which in particular implies 
$$
P(\limsup_{n\to\infty}|S_n−W(n)|/\ln n\le C)=1.\Big)
$$ 
This covers your case, by letting $X_1$ be the standardized version of a Bernoulli r.v. with parameter $p\in(0,1)$.