If you reside in NYS the answer is generally yes. If your child will file a federal return, then you need to file a state return. If you are claiming your child as a dependent and (s)he made more than $3,100 as a single person, then yes. If your child made less than $3,100 you may want to file their taxes in the case that they have a refund or can claim any credits. The form is the same for dependents, the IT-201.
Federal requirements are different. A dependent child must file if (s)he earned more than $12,550. Again, your child may want to file if (s)he earned less than that if there was federal or state withholding that can be refunded.
Remember that you can still claim your child as a dependent even if you claim him or her on your taxes. If you are filing the taxes yourself, make sure that next to "Someone can claim:" you check "You as a dependent". The standard deduction for dependents is the greater of either $1,100 or the earned income plus $350.
