TL:DR Yes, absolutely. At least in aristocratic memory.

Who is Jeremy Corbin?

First of all, the Gracchi were one step in a line of "trouble-making" tribunes. Lily Ross-Taylor wrote an important article "The Forerunners of the Gracchi," in JRS 52 (1962) if you can get it. She challenged the traditional narrative that the Tribunes had been subverted by the Senate and shows there were basically always Tribunes who used the powers of the Tribune to subvert Senatorial control. So they weren't all that strange; and about a generation later you get M. Livius Drusus, who was REALLY radical (and perhaps a direct consequence of the Gracchi's activities, but more on that later).

All the sources report that Tiberius was concerned about the gr owing urban poor and the collection of land among a few very wealthy land holders. Rosenstein at OSU (Rome at War I think is the title of the book) has fairly recently (10 years ago?) challenged this narrative - his argument is that there was no dearth of small farmers; they just didn't want to enlist in the army anymore, but since the question is about the Gracchi's influence, let's stick to the narrative the Romans told themselves. The aristocrats were gathering up huge tracks of land owned by the state meant to be rented in amounts no larger than 500 jugera/citizen (don't ask me to translate that into acres- I am not up to math at the moment and I have no idea anyway). According to Plutarch, Ti.'s plan was to seize public land illegally held and redistribute that land to the urban poor. The poor would have land, the state would have soldiers, and everything would be great.

This required a survey though, to find out who held too much land and to redistribute it in the appropriate amounts to the right people. To this end Ti. passed a law to form a commission of three to conduct the survey etc. after some politicking, but the Senate refused to fund the commission. At this point Attalus, king of Pergamon, died and willed his state and fortune to Rome. Ti. proposed a law to use this money to fund the commission, and after more politicking, got himself lynched by a mob of angry senators. However, the land commission went ahead with a new commissioner.

Gaius, Ti.'s little brother, was a member of the land comcimission, and in adjudicating the redistribution of land managed accidentally to redistribute land that was not Roman public land, but both properly held private land and land belonging to the Allies (this is from Appian, BC 1). The Allies, not being Roman citizens, could not represent themselves in court, and got Scipio Aemilianus to represent them, But he died under mysterious circumstances in 129 BCE. Gaius also ended up getting himself assassinated for politicking shenanigans.

The land problem was left open for about 30 years after C.'s death. Rome's relationship with the Allies turned increasingly sour. in 91 the Tribune M. Livius Drusus tried to pass a law granting the Allies citizenship, and ended up stabbed to death on his front porch. The Allies snapped and fought a very nasty war against Rome which Rome very nearly lost, and only ended when Rome extended (or imposed - the question is still up for debate) the citizenship to (or on) the Allies. That war left Sulla in command of an army, as Consul elect, and directly set up his conflict with Marius for the Pontus command, which in turn led to Sulla's dictatorship and set the pattern for Roman politics until Augustus.

So, yeah, the Gracchi influenced Roman politics. They were not original in using their position to mess with the Senate, but they took it further and forced the Senate further than anyone else, and demonstrated the power the people had to oppose the Senate when organized properly. This proto-communist stuff I'm not so sure about; they didn't seem, to my eyes at least, to have an ideological commitment to social equality, but instead sought to address specific problems (lack of solders, urban poor) and, according to their enemies, use those that benefited from these reforms to further their own careers.

This is the story that all the surviving sources give us, from Cicero to Cassius Dio. Much of this has or is being questioned by modern scholars. [EDIT: a sentence dropped out before my coffee this morning: You don't have to believe the revisionists.] Not everyone believes Rosenstein, for instance. But as far as what Romans themselves thought, yes, the Gracchi did have significant influence on the course of Roman history.

I'd add to what /u/LegalAction said that, on the subject of the Gracchi being "the first socialists," the current balance of opinion is that such a claim is pretty ridiculous, the Gracchi were no such thing. There's a perception for some reason that politicians who appealed to popular support were necessarily in favor of popular reforms, and during the late Republic that's not necessarily true in the least. Gaius Gracchus, for example, actually limited the powers of the tribunate, by preventing tribunes from vetoing the assignment of consular provinces, a law that the OCD points out "shows how far he was from being a 'democrat.'" Tatum points out, which /u/LegalAction sort of mentioned when talking about Tiberius Gracchus' intended replenishment of the military classes, that the Gracchi's reforms were generally decidedly in favor of the status quo, particularly the maintenance of the property classes. In a speech preserved by Gellius (which, genuine or not, is still useful to our point here), Gaius Gracchus declares openly that nobody approaches the Roman people without the hope of getting something from them--in this very speech Gracchus hopes to raise the taxes on Roman citizens! And even the Romans didn't necessarily see the Gracchi as opposed to the aristocracy at all. Plutarch mentions that Gaius Gracchus' land reforms included a provision that the recipients pay rent to the public treasury, something not too different from the current state of affairs, whereas Livius' released them from lease payments entirely. Gaius Gracchus' planned colonies were to be settled by "the most respectable of the citizens" (τοὺς χαριεστάτους τῶν πολιτῶν), a term that brings Cicero's boni to mind--Plutarch places this in opposition to Livius' intention to found twelve colonies and settle them with the urban poor (τῶν ἀπόρων).