Take the Probabilistic Powerdomain of Evaluations as example, the base category is Dcpo, directed complete posets with Scott continuous functions, does the finite product of the Kleisli category of this monad exist?

Thank you. But I am not clear that when and where the limit or colimit of the Kleisli category exists? Another question, if the base category is not cartesian closed, is it possible that the Elienberg-Moore category cartesian closed?