Under Common Law, the master is responsible for the actions of the servant. Respondeat Superior.
If the agent (teacher) is in the course and scope of his or her employment, the employer (school district) would likely be found vicariously liable for the agents misdeeds.
State by State case law might shed light on whether or not a teacher would be considered in the course and scope of his or her employment in terms of the illicit relationship, but, personally, I would argue the teacher wasn't.
The point jespah has made speaks to the direct negligence of the school. If the school had reason to suspect a relationship was in the works and did nothing about it, or if the teacher had a record of such misdeeds and the school maintained his or her employment, then the school might be found liable for whatever damages are claimed, but it would be on the basis of its own tortious act or omission and not on the tortious behavior of the teacher.
A simpler example is one wherein you own a carpet cleaning business. While performing carpet cleaning at a customer's home, one of your employees, negligently knocks over a vases and breaks it. In all likelihood you would be held responsible for the employee's negligence if your customer sued you for her damaged vase.
If, however, it can be established that the carpet cleaner was an independent contractor and not your agent (employee) Respondeat Superior
should not apply.
What makes someone an agent or an independent contractor is fairly straight forward, but the results of litigation are not always straight forward.
The general public likes to think that an employer is responsible for any act of its employee, but that's not the case.