If $\mathcal{C}$ is a small category with pullbacks, a basis for a Grothendieck topology on $\mathcal{C}$ is a function which assigns to each object $C$ a collection $K(C)$ of families of morphisms with codomain $C$ such that:
$(A)$ if $f: C' \to C$ is an isomorphism, then $\{f: C' \to C\} \in K(C)$;
$(B)$ if $\{f_i: C_i \to C\} \in K(C)$, then for any morphism $g: D \to C$ the family of pullbacks $\{\pi_2: C_i \times_C D \to D \mid i \in I\}$ belongs to $K(D)$;
$(C)$ if $\{f_i: C_i \to C\} \in K(C)$ and if for each $i \in I$ one has a family $\{g_{ij}: D_{ij} \to C_i \mid j \in I_i\} \in K(C_i)$, then the family of composites $\{f_i \circ g_{ij}: D_{ij} \to C \mid \, i \in I, j \in I_i\}$ belongs to $K(C)$.
Now, if $K$ is a basis on $\mathcal{C}$, it generates a topology $J$ by
$$ S \in J(C) \iff \exists R \in K(C) \,\, R \subseteq S $$
At this point, as pointed out by Borceux (Handbook of Categorical Algebra, Vol. 3), Grothendieck topologies on $\mathcal{C}$ form a locale, and meet agrees with the usual set-thereotic intersection of Grothendieck topologies. This means that whenever we have a collection $\mathcal{S}$ of sieves on an object $C$, we can construct the smallest Grothendieck topology containing it. But, which is such a closure?
Clearly, we must first add the maximal sieve, if it does not belong to $\mathcal{S}$. But, how to continue such a construction?
Here, I found something:
Description of generated Grothendieck topology
But I don't know if it answers to the general problem.