ANSWER SIMPLIFIED EVEN FURTHER:
Both husband and wife work
This has nothing to do with paying zakaat.
and the wife owns jewellery.
This is where zakaat does come in!
Does the responsibility to pay Zakat fall on the husband.
No, but he can pay for it if he wishes to do so, AND as long as the wife gives him permission to pay for it. If he doesn't want to pay for it, he does not have to, if the wife gives him permission to pay for it, he still doesn't have to pay for it.
(who is earning)
Again, this has nothing to do with zakaat. Zakaat is not based on who is earning and how much is being earnt. It is based on what you have and how much you have had in your possession as accumulated wealth for a complete lunar year.
should the wife pay for her own Zakat
Yes she should, unless her husband offers to pay on her behalf and she gives him permission to pay. If she does not give him permission, he cannot pay for it. Likewise, if she gives him permission and he does not want to (or can't) pay for it, he does not have to. So the responsibility belongs to the wife.
The below fatwa could not be more suited to your question.
ORIGINAL SIMPLIFIED ANSWER:
Everyone pays their own zakat, however, someone can pay on your behalf if they want to for some extra reward from Allah, but no sin on them if they don't, as it's 100% their choice (voluntary) to pay for someone else. Also, zakat is not about who works, it's about the wealth you have had for a complete lunar year after necessary expenses.
Have a look at this fatwa: Source
Praise be to Allaah.
Whoever owns the minimum amount of gold (nisaab),
namely 85 grams, and one full hijri year has passed, is obliged to pay
zakaah on it, by giving one-quarter of one-tenth (2.5%), whether that
is paid from the gold itself or the equivalent after selling it, or
from other wealth.
As you do not own any other wealth with which to pay the zakaah, you
have to give the zakaah from the gold itself, or sell some of it and
pay the zakaah. If someone (your father, brother, husband, etc)
volunteers to pay zakaah on your behalf, that is permissible, and he
will be rewarded for it.
Shaykh Ibn Baaz (may Allaah have mercy on him) said:
If zakaah is owed by a woman who owns jewellery, and her husband or
someone else pays it on her behalf with her permission, there is
nothing wrong with that. Zakaah does not have to be paid from the
jewellery itself, rather it is sufficient to give its value every time
one hijri year passes, according to the value of gold and silver in
the market at the time when the year is completed.
From Fataawa Islamiyyah, 2/85
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) was asked: Is
zakaah due on gold that is prepared for adornment, if a woman does not
have any means of paying zakaah other than selling some of the gold?
The correct scholarly view and the most likely to be correct in my
view is that zakaah must be paid on jewellery if it reaches the nisaab
(minimum threshold), which is eighty-five grams. If it reaches this
amount, zakaah must be paid on it. If she has other wealth and pays
from that, there is nothing wrong with it. If her husband or one of
her relatives pays it on her behalf, there is nothing wrong with that.
If neither of these options is available to her, then she should sell
some of it and pay zakaah with that money.
Some people may say: if we do this, soon she will not have any
We say: that is not correct, because if it falls below the nisaab,
even by a small amount, no zakaah is required. In that case she will
inevitably have some jewellery left.
Majmoo’ Fataawa Ibn ‘Uthaymeen, 18/138
And Allaah knows best.