Yes and no. New York does not tax unemployment from California, but the benefits are included in figuring the overall tax rate @Toon99.
New York will base your child's tax rate on total income (unemployment is taxable in NY, but not California). The actual tax will be the NY income percent of that total tax.
For example, if your child earned $10,000 total and $3,000 in NY, their tax rate would be computed as if they earned $10,000 in NY.
Say the NY tax on $10,000 is $1,000. Then their tax would be 30% ($3,000/$10,000) of $1,000 or $300.
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