If you guys are filing taxes jointly, then you guys are jointly claiming the dependent on your joint tax return. It's not a case of "you claiming" or "her claiming" the dependent.
The point of your W4's is to get the your total federal and state withholdings relatively close to the actual amount of (federal and state, respectively) tax liability for the year, i.e. not underwithhold by so much as to have an underpayment penalty, and not overwithhold by too much as you would just be giving the government an interest-free loan. The things you "claim" on the W4 are simply hints to help you get to approximately the right amount of withholding for your situation, but ultimately what matters is whether your withholding is right for your tax liability for the year based on how you will actually file your taxes, and you want whatever numbers on the W4 will make this happen as accurately as possible.
In a case where there are two jobs, it gets more complicated because if each of you fill out the W4 like it's the only job, you will probably get the wrong withholding. The W4 form suggests for you to use the online tax estimator, or the Multiple Jobs Worksheet, to calculate the right numbers. It also says to claim the dependents and deductions on only one W4. There is a degree of arbitrariness in this, because claiming deductions on one W4 will decrease the withholding in that job, while the other job will have full withholding, or you can switch it around and have deductions claimed on the second W4 and not the first, and the second job will have decreased withholding while the first will have more, or you can split it half-and-half. The end result of these will be similar, though the form suggests that it will be most accurate if you claim deductions on the highest-paying job; but if you know what you are doing then you can adjust the numbers in the "claim deductions in lower-paying job" case to make it work too.
Probably only your job will withhold CT tax, and probably only her job will withhold NY tax, so there is no trading of withholding between two jobs in those cases. Since she is a CT resident with NY income, she will be subject to both states' taxes on that income, but claim a credit on her CT tax. Since NY tax is probably higher than CT tax, the credit will probably equal all of her share of CT tax, so the CT tax will only be only be on your income, but the tax rate will be based on your combined income. And you will effectively only get "your share" of deductions and credits on CT tax, because her share of the tax was credited away, so "her share" of the CT deductions and credits carries no benefit. She will file an NY nonresident return with her NY income (though the tax rate will be based on your combined income), and on there she will only get the "NY fraction" (basically, "her share") of deductions and credits.
So basically, on both CT and NY taxes, you guys will only get a fraction of your deductions and credits, corresponding to your respective shares of the income. This means if you claim the normal reduction in withholding based on your federal deductions and credits, it will probably be too much reduction. So you will have to figure out how to enter a part of it. But in the end, state tax amounts are pretty low, so being a little bit off is probably not going to matter much anyway.
