In English, the prefix occurs mostly in words directly loaned from Greek, such as eulogy, euphemism, evangelist, eucharist, eucalyptus, eudemonic, euonymous, euphoria, eurhythmy, euthanasia, eutrophy, but it has some productivity, combining with Greek loanwords that did not take the prefix in attested Greek, e.g. in eucaryote, eugenic, euchlorine, Eutopia.
Note that the Eu- in Europe is unrelated.
The antonymous prefix is dys- (δυσ-), PIE *dus-, Sanskrit dur-. This prefix appears to be more productive for new coinages, thus dysangelical is a nonce word expressing the opposite of "evangelical", or dyslogy expressing the opposite of eulogy, dysgenic expressing the opposite of eugenic, etc. It also frequently figures in terms for illnesses and ailments, such as dysarthrosis, dyslexia, etc.