Surprisingly, it's estimated that not only do about half pay, but they pay tens of billions every year. And I imagine a formal 'second class citizen' status which gets immigration off your back would draw in quite a few more.

FROM: The Atlantic

Angle #1: Bad SS numbers and the Earnings Suspension File

... Whenever employers send in W-2 forms that have Social Security numbers that don’t match with anyone on record, the agency routes the paperwork to what’s called the Earnings Suspense File ...

... The Earnings Suspense File now contains Social Security tax forms that date back to 1937 and are linked to the taxes that were paid on nearly $1.3 trillion in wages. ... illegal immigration is considered largely responsible for the mushrooming of the file, with undocumented workers paying billions in taxes ...

Angle #2: Individual Tax Identification Numbers

... the Internal Revenue Service has made it easier for workers to pay taxes if they don’t have a social security number (or a fake one, for that matter). Workers who are paid illegally in cash can still pay their taxes with an Individual Tax Identification Number (ITIN), filing a return just like any other taxpayer; having a history of paying taxes can be an important step in securing legal status.

Revenue figures:

• [ITIN] In 2010, about 3 million people paid over $870 million in income taxes using an ITIN, and according to the IRS, ITIN filers pay $9 billion in payroll taxes annually.
  
• [Local & State] The Institute on Taxation and Economic Policy, a Washington, D.C., think tank, estimates that about half of undocumented workers in the United States pay income taxes. They also help fund public schools and local government services by paying sales and property taxes like any other resident. This added up to about $10.6 billion in state and local taxes in 2010, according to an analysis by the institute.
  
• [Bad SS numbers] **the Social Security Administration, estimates that about 1.8 million immigrants were working with fake or stolen Social Security cards in 2010, [paying] $13 billion into the retirement trust fund that year, and only got about $1 billion in benefits.
  

Via local, state and federal taxes (via ITIN and bad SS numbers), we're already making tens of billions off them per year.