Is it permissible for a husband to give his zakaah to his wife because she has debts?

Yes, there is nothing wrong with that, because he is not
obliged to pay off her debts. What is not allowed is for a man to give his
zakaah to his wife in order to waive the obligation of spending on her
maintenance. 

Ibn al-Mundhir (may Allah have mercy on him) said:

The scholars are unanimously agreed that a man should not
give his wife zakaah so that she will not need any maintenance after that,
because spending on her maintenance is obligatory for him, so it is not
permissible to give zakaah to her, as also applies to giving zakaah to her
for her maintenance.

End quote from al-Mughni. 2/270 

‘Ulaysh (may Allah have mercy on him) said in Minah
al-Jaleel Sharh Mukhtasar Khaleel:

With regard to the husband giving his zakaah to his wife,
that is not allowed according to scholarly consensus. That [i.e., either
spouse giving zakaah to the other] is not allowed unless it is given by one
of them to the other in order to pay off debts or spend on someone else; in
that case it is permissible according to consensus. End quote. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: If
a person pays off his father’s debt from his zakaah, and his father was
unable to pay it off, then that is permissible. If a husband gives his
zakaah to pay off his wife’s debt, and she is not able to pay it off, then
there is nothing wrong with that, because he is giving charity to them on
the basis that they are in debt.

End quote from Fataawa Noor ‘ala ad-Darb. 

To sum up: it is permissible for a husband to give his zakaah
to his wife to pay off her debt. 

And Allah knows best.