You need to withhold the employee's portion of FICA/SSA (employment taxes). You are not required to withhold income taxes; however, many employers do choose to withhold because it is much more convenient for the employee. (If income taxes are not withheld, the nanny must pay quarterly estimated taxes or get hit by a significant penalty by the IRS.)

The information you need (and much more similarly useful info about being a household employer) is contained in IRS Publication No. 926, which is available for free on the IRS website.

Possibly dumb question, but if I employ a nanny legally do I need to do tax withholding or can they be responsible for it?