What are my liabilities if I sponsor a student who may remain here beyond the time that she is legally entitled to from her home

You probably would not face liability, but consider the following cautionary tale. I represented a young woman, a lawful permanent resident who sponsored her sister to come to study in the U.S. The sister dropped out of school, continued living with the sister and eventually moved to another state. The Immigration and Customs Enforcement (ICE) Fugitive Operations Team approached my client to determine her sister's whereabouts, which she was unsure of. They gave her a very hard time, appearing at her home early in the morning, coming to her workplace, and putting a hold on her naturalization application. It finally got worked out with ICE, but it was not pleasant for a while. You need to have confidence that the student will comply with all terms and conditions of the visa, and, in my opinion, refuse to sponsor him or her if you do not trust the student. You would not want to endure accusations like the ones my client received, that she was suspected of smuggling and harboring an unauthorized "alien" in the U.S.