My daughter is a college student in New York state, but has an internship in Texas. Is she still liable for taxes in New York?

Money she earns while living and working in TX should not be taxed by NY. It would be most helpful if the address her employer uses is not in NY - otherwise NY will try to claim she owes tax in NY. Does she have an apartment in TX? If so, it should be easy to document that she did not live in NY State during her employment. Also, it is helpful if she is not on a lease in NY during the internship (again to support the claim that she is not a NY resident).

It is NOT clear in which state the daughter lives while not attending school.

If she lived in NY state before attending college there, then NY state would be considered to be her HOME state, while the internship in Texas is a temporary job.

This being the case, NY state does, In my opinion, have a claim on the income earned while working in Texas.