Because Social Security's benefit formula is designed to replace a larger share of pre-retirement income for lower-income workers than for middle- and upper-income workers, it would seem to help poorer Americans.
However, many younger low-income workers would have even more retirement income if they were instead able to invest their Social Security taxes in safe government bonds or broad stock index funds. Furthermore, Social Security's high tax rates (10.6% of wages) prevent low-income workers from accumulating private savings.
This "crowding out" of private savings by Social Security taxes also makes it much more difficult for the poor to improve their lot, such as by purchasing a home, starting a business or investing in bonds, stocks or real estate.
Perhaps most importantly, the Social Security System hinders the ability of low-income workers to pass on wealth to the next generation.
Because Social Security is a "defined benefit" pension program, the payments end when the beneficiary dies. For those who die at a ripe old age, it can be a fairly good deal. But for those who die before, or soon after, retirement the benefits are much less than the taxes paid. However, unlike privately held investments, the difference can't be passed on to heirs.
This means that the children of the poor have a higher probability of remaining poor than they otherwise would if their parents had been able to invest their Social Security taxes. For example, today the children of parents with less than $99,000 of bequeathable wealth at retirement have a 40% probability of themselves reaching retirement with less than $99,000 of bequeathable wealth. In contrast, if Social Security didn't exist (or were privatized) those children would have only a 16% chance of retiring in the poorest group of their generation and, conversely, an 84% chance of retiring better off than their parents.1
1 Gokhale, Jagadeesh, "The Impact of Social Security Reform on Low-Income Workers," The Cato Institute, SSP No. 23, December 6, 2001, pp. 4-6.This paper can be found at: http://www.socialsecurity.org/pubs/ssps/ssp23.pdf.