Is it permissible for a veiled wife to work in the business of household utensils, where there is a strange man assisting her, provided that she only deals with women and the man only deals with men?

If the wife's work exposes her to being alone with non-mahram men (khalwa) or to mixing that incites temptation (fitna), then you are not permitted to allow her. However, if the work does not expose her to that, and she adheres to the Sharia's regulations in dealing with non-mahram men, such as avoiding a soft voice, limiting speech to what is necessary, observing the legal hijab, and temptation is secure, then there is no objection. It is preferable for a woman to remain in her home, but if she needs to work, it should be in a place where she does not mix with men.