This is an odd question. Immigrant children are just like any other children. In light of this, the government has the responsibility to educate them up to a certain level. Through taxes, the government provides basic education until college. So, this also applies to immigrant children. In other words, if a person pays taxes and is in this country legally, then they have all the rights of a citizen.
If you are speaking of illegal immigrants, then this complicates things considerably. The nation is divided, to say the least. Some say that no government benefit should be given to illegal immigrants, which would include education for their children. Others point to the hard work of immigrants and say that education should be given to them. In the end, it is a very controversial point of contention and there is no easy answer. I suspect, this will be one of the debating point for the next election.