As detailed in the attached memo (PDF) from the State Education Department's Senior Deputy Commissioner John B. King, Jr., "school districts may not deny resident students a free public education on the basis of their immigration status, as long as they meet the age and residency requirements established by state law."
In a 1982 decision, the U.S. Supreme Court recognized that children who are undocumented immigrants cannot be denied a free public education if they are, as a factual matter, district residents. Accordingly, at the time of registration, schools should avoid asking questions related to immigration status or that may reveal a child’s immigration status, such as asking for a Social Security number.
Please note that the department does not require districts to collect students' social security numbers for any purpose.
The complete memo (PDF) is available for download.