for every tuple $(a \in C,b \in C)$ an object $C(a,b) \in \mathcal{V}$ “of morphisms” from $a$ to $b$ in $C$;

for each sequence $(a_i \in C)_{i = 0}^n$ of objects in $C$ an object $C(a_0, \cdots, a_n) \in \mathcal{V}$ “of sequences of composable morphisms and their composites”;

such that these composites exist essentially uniquely and satisfy associativity in a coherent fashion.

One way to make this precise in a general abstract way should be to define $C$ to be a ∞-algebra over an (∞,1)-operad in $\mathcal{V}^{\otimes}$ over Assoc${}_{Obj(C)}$, the $Obj(C)$-colored version of the associative operad;

Once a model category $V$ for $\mathcal{V}$ has been chosen, one can consider semi-strict$\infty$-enrichments given by ordinary $V$-enriched categories equipped with a notion of weak equivalence that remembers that these are presentations for enriched $(\infty,1)$-categories. See also enriched homotopical category.

More generally, for $R$ an E-∞ ring then an $R$-linear (∞,1)-category? is naturally enriched in $R$-∞-modules. (This includes the previous case for $R$ the sphere spectrum.)