Note: I have edited this answer further because I was being silly before (unnecessarily restrictive).
I take it you mean categories $E$ for which $- \times -: E \times E \to E$ preserves quotients. The word 'quotient' may be slightly ambiguous because sometimes people use it to mean'coequalizer', and sometimes just 'epi' (as in 'quotient object'), but I take it you mean 'coequalizer'.
A reasonably large class would be regular cartesian closed categories, which includes categories of algebras of monads on $Set$ and semi-abelian categories and toposes. The key thing here is that Here quotients = regular epis are stable under pullback and in particular are closed under taking products on either side. Furthermore, in a regular category, every quotient is a reflexive coequalizer, meaning a coequalizer of a pair $f, g: X \to Y$ for which there exists $h: Y \to X$ with $f \circ h = g \circ h = 1_Y$. The In particular, the two projections $\pi_1, \pi_2: E \to Y$ of an equivalence relation $E$ on $X$ X$, for example the kernel pair of a quotient, form a reflexive pair by the reflexivity property, so . So in a regular category, where quotients = coequalizers are necessarily coequalizers of their kernel pairs, quotients are quotients of reflexive pairs.
The reason this reflexivity is relevant is a $3 \times 3$ lemma which says that in a (edit: commutative-in-parallel) diagram of $3 \times 3$ objects in which all rows and all columns are coequalizer diagrams of reflexive pairs, the diagonal is a coequalizer diagram. See the first page of Johnstone's Topos Theory. Then apply this lemma to the evident diagram whose rows are of the form
$$X_i \times X_{1}' \stackrel{\to}{\to} X_i \times X_{2}' \to X_i \times X_{3}'$$
and whose columns are of the form
$$X_1 \times X_{j}' \stackrel{\to}{\to} X_2 \times X_{j}' \to X_3 \times X_{j}'$$
In the category $Top$, it would therefore be natural to consider quotients by equivalence relations (or even just reflexive relations) which are preserved by taking products on each side. It's that latter condition which needs to be characterized (or at least discussed further), and I may come back to that later after I get the kids off to school. :-)
Edit: For a discussion of topological quotients which are stable under taking a product on either side, see the paper by Day and Kelly, On topological quotient maps preserved by pullback or product, Math. Proc. Cam. Phil. Soc. 67 (1970), 553-558. Or google Day-Kelly maps to find out more.
