From an economic standpoint, it's a consensus among economists that allowing for the liberalized movement of capital and goods is an overall economic benefit. In a free society, if we strive to have goods and capital move freely, that same principle should apply to labor, as well. In this case, the economic case for further legalized immigration is solid. This would mean doing so in its purest form. We encourage labor movement in an interstate fashion, so why is it suddenly a problem when we apply that to the international level? Furthermore, restricting immigration curtails the freedom of movement for millions across the planet.
Let's consider the costs, not to mention the level of government intervention, to enforce our borders. How about the $5.5B per annum we spend on Immigration Customs and Enforcement alone? Let's not forget other costs, such as people dying when crossing the border, the fact that the United States' Cold War foreign policy attributed to the conditions that encourage migrants to flow to the United States in the first place, border control spending, a larger underground economy, perpetuating the inaccurate stereotype that undocumented workers are nothing more than moochers, as well as corrections spending to incarcerate those we label "illegal immigrants."
It's not as if there isn't precedent for the idea of implementing open borders. The European Union de facto allows open borders for member countries through the Schengen Agreement. There are similar agreements between varying countries, including Australia and New Zealand, India and Nepal, Ireland and the UK, and Belarus and Russia. Not only that, having very open borders was American immigration policy for much of the 19th century.
I think the argument for open borders is a strong one (see here, here, here and here). Unfortunately, most Americans don't see the importance or need for more immigration. As a matter of fact, a plurality would like to see immigration decreased instead of increased. The Overton window, or the idea that only politically feasible ideas are the best ones to pitch, is the single largest hurdle from ever making open borders, or even more open borders, a reality. Even if you want to factor in national security concerns and have at least some restrictions, the trend should be towards immigration liberalization, not away from it. Since closed borders denies the freedom of movement, one of the most basic freedoms out there, the burden of proof should go to those against open borders, especially when considering the billions of dollars the global economy loses out on as a result (Clemens, 2011). In a polemic environment, that is how burden of proof works, but I know it's far from the case when dealing with the political process. The best chance that America has for having anything resembling open borders is to keep advancing the economic and moral arguments, and hope that Americans can change their minds on the issue.
10-20-2014 Addendum: See this good argument for a libertarian case for open borders.
11-17 2015 Addendum: An interesting deontological, libertarian argument against open borders and how open borders are actually an assault on private property.