Because it's so popular and a so-called "third rail" in politics Social Security survives largely intact although it pays out less in inflation adjusted benefits than it should, but Democrats have their sights set on it. Their current leader, President Obama, when he took office formed a blue ribbon commission to give him debt reduction recommendations and packed it with Democrats and Republicans who want to privatize Social Security. (Democrats like Erskine Bowles, John Spratt and Dick Durbin, Republicans like Allan Simpson and Paul Ryan) and although that attempt was turned back he still makes ominous references to our "long term debt" -- code for Social Security -- despite Social Security not being part of the federal budget.
Democrats refuse to refute assertions that Social Security if part of the federal budget or the constant claims that the program is going broke despite the fact that it's been shown over and over that it's not and that a slight adjustment to the "cap" would beef it up substantially; only the first $118,500 of a taxpayer's income is subject to Social Security contribution deductions, so millionaires and billionaires basically get off scot free and yet still draw their benefits.
New Democrats like those we have in New Mexico now usually profess support for the program when pressed but if you read statements they make about it it's always a qualified support, or they simply avoid the topic altogether. None has come out with a robust endorsement of the program as is. On their web site you'll see statements about the listing of the Lesser Prairie Chicken as a threatened species but not about Social Security as a threatened program that millions of retirees depend on.
Republicans have never liked Social Security (although as they do with all other government benefits they complain about they take it) and make periodic attempts to privatize or undermine it. Democrats declining support for it must be seen in the context of their rightward transformation in the past 40 years and adoption of Reaganomics economic policies -- see the post below.