I assume that by "Social Security" the question means retirement benefits. Those benefits and Medicare are very different programs, even though they are both run by the SSA, so I'll discuss them separately.
The design of Social Security retirement benefits is predicated on the assumption that people are unwilling or unable to save for retirement themselves. Therefore, the argument goes, the government should force people to save through payroll and self-employment taxes, and use that money to provide pensions.
So, one fundamental question is: is that assumption correct? Unfortunately, the answer is yes. This is not even just due to some cultural American spendthrift attitude - it seems to be true around the world, and a big part of the reason for high birthrates in many places is to ensure the existence of children who will take care of the parents after they are no longer able to work.
The next fundamental question is: should we care? Does it matter that people who neither save enough nor raise children who will support them might wind up spending their "golden" years homeless? This is where the ideological divide seems to be at its most stark. Liberals feel, with some justification, that everyone who has worked (or, through marriage, supported a worker) for years deserves enough income to live with dignity for their final years. Conservatives feel, with some justification, that the maintenance of a massive government bureaucracy only serves to save people from having to take personal responsibility.
In this case, the liberal argument is a stronger one. This is partly due to the realities of retirement planning. For most of the population, what is really needed for a secure retirement is an inflation-adjusted pension with spousal survivor benefits. Buying such a pension from a private insurance company is prohibitively expensive, largely because inflation is so unpredictable. For a typical worker making a 5-figure income, they would have to set aside roughly half their income during their working years to afford such a pension.
So, yes, SS retirement benefits are a worthwhile program. It's the only way that most people will ever be assured of a retirement income. The program certainly has problems as designed: it tries to support too long of a retirement period, it is not sustainable with current demographics, etc. But such problems could be fixed and don't remove the inherent value in such a program.
(For brevity, I am not going to discuss SS disability benefits in depth here. However, I think that is also a worthwhile program, although I think it needs to be made more resistant to abuse.)
Medicare is a government-run health insurance program. There are two main questions here:
1) Should the government ensure that retirees (and the disabled) have health insurance?
2) Should the government be running a health insurance program?
The first question is similar to the questions around retirement benefits. Health insurance would be unaffordable to many retirees, and there's realistically no way that most workers could ever plan to have private health insurance. Given that society has an interest in not letting people suffer from untreated illnesses, it is worthwhile for government to make sure that people past their working years can still get health treatments when necessary.
The second question is much more difficult. Medicare works only because the federal government can mandate it to work. In many cases, Medicare reimbursement rates are much lower than the cost of providing a service, meaning that medical providers often lose money on Medicare patients, subsidizing them by charging more to private patients. Bureaucrats at the Centers for Medicare and Medicaid Services put out reams of regulations on procedures designed to reduce utilization of costly procedures, and then bureaucrats at the Dept. of Health and Human Services' Office of the Inspector General put out reports criticizing the regulations. Then Congress overrides much of this by legislation anyway. The result is a horrible, inefficient system that wastes much more money than any private health insurer, and which only appears to work by making health care more expensive for everyone else.
So, Medicare is worthwhile in intent, but destructive in implementation. We would be much better off insuring seniors as part of a universal coverage program that insured everyone else too. Such a universal program would stop the huge distortions created by Medicare and spread the costs of policy-making over a larger base. (I do not personally favor such a universal program being run as a government program as Medicare is, for the reasons I just described. But a universal government plan would, of necessity, need to work more fairly than Medicare does.)
Therefore, I would say that Medicare is not worthwhile. While insuring retirees is important, Medicare does it badly and at great (hidden) cost to everyone else, distorting access to healthcare. Note that Obamacare in no way changes this.
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