(Since you are looking for a reference, I turn my comment above into an answer:)

A proof using classical fluctuation theory is given my answer to 

https://mathoverflow.net/questions/147270/expected-supremum-of-average

(I'm not aware that this result is well known, or of earlier references).

ADDED: 

Consider the associated Poisson process $N(t)$ with $N(0)=0$ and interarrival times $X_i$. Then is is easy to see
that for $a>0$
\begin{align*}
\sup_{t\geq 0}( N(t)-at) \leq 0 \;\; \Longleftrightarrow \;\;\inf_{n\geq 1}\frac{S_n}{n}\geq \frac{1}{a}\end{align*}

It was shown here https://www.ams.org/journals/tran/1957-085-01/S0002-9947-1957-0084900-X/S0002-9947-1957-0084900-X.pdf and here  https://www.jstor.org/stable/2237099 that 
\begin{align*}\mathbb{P}(\sup_{t\geq 0} (N(t)-at)\leq 0)=\Big\{\begin{array}{cc} 1-\frac{1}{a} \mbox {  if  } a\geq 1\\
                                                                       0 \mbox{ else }\end{array}\end{align*}

Thus in this formulation the result is indeed classical.