Reference for "It is enough to specify a sheaf on a basis"? The wikipedia article on sheaves says:

It can be shown that to specify a sheaf, it is enough to specify its restriction to the open sets of a basis for the topology of the underlying space. Moreover, it can also be shown that it is enough to verify the sheaf axioms above relative to the open sets of a covering. Thus a sheaf can often be defined by giving its values on the open sets of a basis, and verifying the sheaf axioms relative to the basis.

However, it does not cite a specific reference for this statement. Does there exist a rigorous proof for this statement in the literature?

This is an excellent question and to tell the truth it is often handled in a cavalier fashion in the literature. This is a pity because it is a fundamental concept in algebraic geometry.  
For example  the structural sheaf $\mathcal O_X$ of an affine scheme $X=Spec(A)$ is defined by saying that over a basic open set $D(f)\subset X \;(f\in A)$ its value is $\Gamma(D(f),\mathcal O_X)=A_f$ and then relying on the mechanism of sheaves on a basis to extend this to a sheaf on $X$.
The same procedure is also  followed in defining the quasi-coherent sheaf of modules $\tilde M$ on $X$ associated to the $A$-module $M$.
However there are happy exceptions on the net , like Lucien Szpiro's  notes where sheaves on a basis of open sets are discussed in detail on pages 14-16.
You can also find a careful treatment in De Jong and collaborators' Stack Project , Chapter 6 "Sheaves on Spaces", section 30,  "Bases and sheaves"
