I've been looking for a full transcript of Giuliani's 1996 speech to the Kennedy School of Government that has generated the recent YouTube, and the closest I could find was this, which as far as I can tell, is a partial transcript of the speech. It doesn't contain the section that was featured on the YouTube clip, but I do think it provides a fuller picture of Giuliani's thinking at the time. And I don't think he comes across as badly. 

Giuliani said:

Illegal immigration is a different matter. I do not defend it. No one should break the law. But preventing illegal immigration is the job of the federal government. The United States has to do a lot better job of patrolling our borders. If we can't stop illegal immigration, then we can't stop drugs and weapons from entering the country, either.

But in a country as large as ours, with our protection of individual liberty, and with a huge border that spans sea, deserts and mountains, and given the strong desire that people have to come to this country, the federal government may never be able to stop illegal immigration completely. At best, all we can expect is that the federal government will do a better job of patrolling our borders.

The reality is, people will always get in. And the reality is, the federal government does not deport them. In New York City, which has 400,000 undocumented immigrants, only about 1,500 a year are deported. Under the new federal legislation, that number would — at most — double to about 3,000 out of 400,000.

So illegal and undocumented immigrants are going to remain, and even increase. And nothing that is now being proposed in Washington would realistically change that very much.