Yes, the 14th Amendment's citizenship clause grants U.S. citizenship by birthright to the children of illegal immigrants, because the child only needs to be born here in order to automatically be a citizen. It does not matter if the parent is here illegally. Tourism for citizenship is common for the United States.
The 14th Amendment does grant rights to the children of illegal immigrants that are born in the United States of America. I do think though that it is a policy that is wrong and should be changed. I do not think that the children of illegals should have the right to be American citizens.
Not only does the 14th Amendment clearly guarantee citizenship for children of illegal immigrants, so does well over a centuries worth of judicial decisions.
It doesn't matter whether or not you agree with this. The law is clear and the only possible way to change this would be to amend the Constitution.
The Civil Rights Act of 1866 first issued this right, but the Citizenship Clause of the Fourteenth Amendment reiterates it. If a person is born in the United States, they are a United States citizen. Their parents status has little to do with it, and it essentially means nothing if their parents happen to be illegal immigrants. This is precisely why people crossed the border to have their children here.