Can my wife use her old passport with a valid visa to exit and enter Thailand with a new passport?

This topic actually comes up a LOT

At immigrations they don't transfer visas or entry stamps, only extensions of stays..

So in this case the wife of the O/P would carry BOTH passports when they exit thailand, because even though they have a new passport the visa for thailand in the old one is still valid.

They'll get stamped out on the new passport and then when they re-entry the country they show both passports again and they'll get a 90 day entry stamp from the Non-B in the old passport along with a notation they had a visa in a previous passport

You only go to the immigration office to get extension stamps transferred when you get a new passport (UNLESS you have an ELITE visa which is totally different)