In the grand scheme of things, not much. Let me be honest, I love the Shinsengumi, and as stereotypical as it sounds, they're 100 percent responsible for my getting interested in Japanese history in the first place. But being famous as characters in thousand novels, movies, tv shows, games, musicals, manga, and anime today doesn't mean they had a huge importance to the way the period unfolded. They were threads in an unraveling tapestry with a million threads. The Shinsengumi were always very involved in whatever was going on, but they weren't leading/controlling/characterizing these larger events. So, for example, they were very active and effective in the larger trend of the shogunate and allies solidifying control of Kyoto in the mid 1860s.
I answered this question: At the time, just how famous were the Shinsengumi?, which goes over what they achieved, focusing on their one big claim to influencing the course of events: the Ikedaya Affair. When people have tried to make an argument for the Shinsengumi changing the course of events, they focus on the Ikedaya Affair. Quoting myself:
The Ikedaya Affair, on July 8, 1864 is an iconic moment in the Bakumatsu period. Acting on the information of its spies, the Shinsengumi took prisoner a shop owner named Furutaka Shuntaro, who was affiliated with anti-shogunate activists in Kyoto. Under interrogation (Furutaka was certainly tortured but any details on the internet come from fiction), Furutaka confessed to being part of a plan, in coalition with Choshu domain samurai, to set Kyoto on fire, and use the distraction to kidnap the Emperor. From his confession and other information, the Shinsengumi concluded that the conspirators would be meeting late that night at one of two inns in Kyoto. They notified Aizu domain of the operation, and when Aizu delayed sending back-up, the Shinsengumi split their force of thirty men into two groups to go to the two inns.
The smaller group of eleven samurai, led by Shinsengumi chief Kondo Isami, raided the inn, the Ikedaya, where the meeting was taking place. The conspirators fought back. In the fighting that followed, three Shinsengumi members were killed, and seven of the activists. Reinforcements from the other Shinsengumi group arrived after a while, and the Aizu and Kuwana domain troops surrounded the inn. Twenty-three of the activists were then taken prisoner. Some of the dead and prisoners were samurai of Choshu domain.
This was, without a doubt, the crowning moment of glory for the Shinsengumi. They received official thanks, reward money, and a more respected role in Kyoto law enforcement. The Ikedaya Affair also had a strong effect on the Choshu militia troops who were already raring to fight Aizu and Satsuma (at that time, the two domains were allied) in Kyoto. The Ikedaya Affair wasn’t responsible for putting Choshu on course to invade Kyoto, but it definitely helped take the brakes off that force. On July 19, Choshu forces entered the city with the intention of taking control of the Emperor. In the ensuing battle, known as the Kinmon no Hen or Hamaguri Rebellion, Choshu was defeated by Aizu and Satsuma forces, with the Shinsengumi fighting alongside Aizu’s troops. To cover their retreat, the Choshu troops set fire to the city, so ironically, the very scenario the Shinsengumi had set out to avoid came to pass.
Nagakura Shinpachi, one of the Shinsengumi's few surviving captains, apparently claimed that the Shinsengumi's actions at the Ikedaya had delayed the shogunate's fall significantly. (I say "apparently" because Nagakura's version of events did go through some newspaper rewriting, nothing about the Shinsengumi is ever very sure.) This tack is taken up by a lot of people writing about the Shinsengumi but I find it very doubtful. As I mentioned above, the Ikedaya Affair enraged the Choshu militias, but Choshu troops and Choshu-allied nobles were already expelled from Kyoto, Choshu was on a collision course with the Shogunate/Satsuma/Aizu and already at war with the Western powers along the Shimonoseki Strait. They weren't on the verge of victory and thwarted by the Shinsengumi. They were in spectacular self-destruction mode.
We know now that Choshu would eventually snatch victory from the jaws of defeat, turn their defeat by the Western Allies into a strong relationship with British arms merchants and diplomats, bind together their fractious factions into one force, mend their emnity with Satsuma domain, and go on to take control of the Japanese army for two generations. But reading that future into 1864 is a mistake. Saying, as some popular "historians" like to, that the Shinsengumi delayed the Meiji Restoration by a year, imagines that the Meiji Restoration was a one-way process that was underway in 1864, and anything that happened to disrupt Choshu must therefore have delayed it. History doesn't work like that.
Of course, there's the opposite argument, that the Ikedaya Affair hastened the Meiji Restoration, by further enraging and solidifying Choshu opposition to the shogunate. I'm skeptical about that too. As just mentioned, there were so many variables in Choshu's fall and rebirth. The Ikedaya Affair and the Shinsengumi should just be noted as yet another reason the Choshu leaders harboured a lot of resentment, and a factor in the vicious emnity between Choshu and the Shinsengumi's patron: Aizu, that culminated in the most brutal part of the Boshin War: the Battle of Aizu.
No one doubts that Kondo Isami and Hijikata Toshizo were great leaders, and despite being part of a hopeless, doomed struggle, Hijikata proved to be a great military commander in the Boshin War. It's always fun to imagine an alternate history in which the shogunate pulled itself together and succeeded in their own projects to modernize/centralize Japan, and in that story, you can imagine Kondo and Hijikata serving successfully, but that's the realm of fiction, not history. In the real world, they struggled, failed, and were not necessarily the most influential group of people.