The non-custodial parent does not need the dependent's SSN for child support, as that is not a deductible expense on the federal tax return. HOWEVER, there could be other legal reasons they would need it, and the custodial parent could run in to legal issues (not only with the IRS) if they don't provide it. So you need to ask why they need the SSN. If they have a legal need (such as on a state tax return) then you are legally obligated and required to provide it.
