As a sort-of an anwer (or rather summary what the dispute is about)... Biden gave a rather unclear speech, from which a portion (apparently praising Paul Ryan's approach to Social Security cuts) was reproduced by people close to Sanders.
Biden's campaign says he was essentially quoted out of context, and that in the full speech he is about protecting social security. Thus sites like Politifact have rated Sanders' campaign claim as false.
If there's one para in the press that summarizes the gist of this controversy reasonably well...
More recently, there is Biden’s appearance in 2018 at the Brookings Institution, when the former vice president made a muddled statement that seemed to embrace means-testing Social Security. “I don’t know a whole lot of people in the top one-tenth of one percent, or the top one percent who are relying on Social Security when they retire,” he said, before going on to discuss his own tax plans, adding that it (and Medicare) “still needs adjustments.” Biden now says he was being sarcastic, and is backed up by PolitiFact, but left-wing commenters remain skeptical of the explanation.
The "left-wing commenters" that that alludes to are other journalists (e.g. a piece in The Week) saying that what Biden means by "protecting" is actually making some cuts in order to save the programs as a whole. This is based on prior speeches of Biden, since in the one in question it's not clear what he means by "protecting". It seems to me however that all these prior instances were at least a decade older, so the "connect the dots" exercise is rather tennous if one wants to infer what Biden's position actually is now (as opposed to accusing him of harboring hidden agendas, or not acknowledging his past positions...) It's too long to go over the whole argumentation of coded language here, but the conclusion that The Week draws is:
The Biden campaign claimed to Sherman [Politifact] that "he was mocking Ryan," and Sherman notes that Biden had said many times that he wanted to protect these programs. Therefore he didn't endorse any cuts.
This is not true, but you need to be well-versed in Beltwayese to understand what Biden means by "do something" and "adjustments." For decades, Social Security and Medicare have been the bugbears of centrist austerians who (supposedly) want to cut the federal budget deficit. [...]
So Biden was mocking Ryan for trying to cannibalize Social Security and Medicare to pay for his tax cuts, but in the process Biden endorsed his usual line that we need to cut these programs at least somewhat to make them solvent over the long term and keep down the deficit. "I don't know a whole lot of people in the top one-tenth of 1 percent or the top 1 percent who are relying on Social Security when they retire" clearly implies that benefits for those people could be cut without harming them. Saying that Biden lauded Ryan for doing so is arguably an exaggeration, but he absolutely did say cuts should happen.
Politifact simply doesn't know what it is talking about — it is Sherman who did not consider the relevant context.
One article accusing Biden of duplicity on the matter does bring up a more recent issue of the "chained CPI" from the Obama era, which supposedly Biden supported, but I lack the technical knowledge on the matter to say whether it amounted to cuts or not (the article claims it does).
On the other hand, Biden's 2019 plan calls for an increase in Social Security.
It's hard to select a small-enough nutshell of how Krugman has analyzed the issue, but I'd go for:
Not to sugarcoat it: The Sanders campaign has flat-out lied about things Biden said in 2018 about Social Security, and it has refused to admit the falsehood. [...]
That said — and this is no excuse for the Sanders camp — it would be good to have Biden explain why, in the more distant past, he went along with the Beltway consensus that Social Security needed to be pared back. [...]
a Sanders adviser recently circulated a snippet from the video of the event that made it appear that Biden was actually supporting Ryan’s position and calling for Social Security cuts. [...]
Biden did make a misstep in his counterattack, mislabeling the misrepresented video clip as “doctored,” but that doesn’t mean he’s not still due an abject apology. [...]
So, about the element of truth in the criticism of Biden: Once upon a time, there was a peculiar consensus among media figures and would-be centrists that the long-run cost of entitlement programs was America’s biggest problem, that Social Security in particular was in crisis and that something had to be done, with the solution including benefit cuts. [...]
But that was then. These days, Biden, like many Democrats, is calling for an expansion of Social Security benefits. That doesn’t make his previous statements irrelevant; he should acknowledge that he has changed his position, and his history on the issue is one reason progressives worry that, if elected, he might fritter away his political capital in vain attempts to reach bipartisan compromise.
As you can see various commentators come to somewhat different conclusions. So the whole dispute is based on interpreting (several) layers of language in speeches, some "connect the dots"... and possibly some technical budget matters (which on their own might make a better Skeptics question.)
As for the videos... the original Biden talk is on C-SPAN. The segment that was quoted/debated is around 24 minutes in. The 20-second clip that Sanders' campaign apparently shared is found on Twitter
If we restrict ourselves to the clip "doctoring" issue, CBS has commented that:
It is worth noting that despite Biden's repeated use of the words "doctored" and "fake," the video is neither of those things. While the former vice president's comments were taken completely out of context and the Sanders campaign's framing of his words is misleading, the 20-second-video itself is real and unaltered.