Even if a person is in favor of open borders, in a welfare state the reality is that state budgets are overwhelmed if too many people use social services and don't pay enough into the system to cover the cost. If most of the immigrants are working and paying taxes, then it would likely be a wash, but what happens when there just aren't enough jobs, and the immigrants have families to support?
I don't like the idea of shutting down the border, but the current situation is unsustainable for the states at the border. A strong economy will help, but it might be years before the economy recovers, and by then states might go bankrupt. Our laws and our system of welfare ensure that the problem will get worse.
If we had a truly free market, then there would be less need to worry about immigration -- there's plenty of room and if the obstacles to economic growth are removed, then the extra workers will be needed, and immigrants can then enjoy American freedom and pay their way -- who would have a problem with that? Unfortunately our statist system is not set up to receive many immigrants, unless they are highly educated and trained to do specialized work, without draining state budgets.