This is a matter of expanding the definition, in this case Definition II.1.3 in SGA 4, which defines pretopologies.
By a “base” in this answer I mean what appears to be the most common definition: a collection of subsets of a fixed set $A$ such that any finite intersection of elements in the base is a union of elements in the base.

Axiom PT0 for pretopologies says that any morphism in a covering family admits base changes.
In the category of open inclusions, cobase changes are given by intersections, so they always exist.
(If one restricts instead to the category of open inclusions of sets that belong to the base, then cobase changes exist if and only if for any inclusions $U→X$, $V→X$ of opens in a base, the intersection $U\cap V$ also belongs to the base.)

Axiom PT1 says that pullbacks of covering families are covering families.
This is always true because of the previous axiom and the fact that unions commute with intersection with a fixed open subset.

Axiom PT2 says that covering families can be composed.
This is trivially true for the case under consideration.

Axiom PT3 says that the singleton family consisting of the identity map is a covering family, which is tautologically true in our case.

Thus, a base for a topological space is a pretopology if and only if it is closed under intersection of pairs.

In particular, open balls in a metric space as a coverage on the category of open subsets trivially form a pretopology.
(If we consider the category of inclusions of open balls themselves without any other open subsets, it does not have cobase changes, so open balls do not form a pretopology.)