Does a wife with no income have to file taxes with her husband?

It depends on how the wife is categorized and what assets are in her name, solely or jointly.

Let's say there is a house in both of the names then yes, the wife has to file in a joint return with the husband. If the wife is a student then she will have to file and the husband may be able to claim her as a dependent.

To be safe it is always better to file either a zero return solely or as the spouse on a joint return.