According to the DS-11 passport application form, an SSN is not needed if you don't have one.
Section 6039E of the Internal Revenue Code (26 U.S.C. 6039E) and 22 U.S.C 2714a(f) require you to provide your Social Security number (SSN), if you have one, when you apply for or renew a U.S. passport. If you have never been issued a SSN, you must enter zeros in box #5 of this form.