While the ships of the Imperial Russian Navy played a very small part in the events of 1917 and onwards, their sailors and officers played a significant one. The proximity of the naval fortress of Kronstadt to Petrograd, and the radicalness of the sailors there was a major factor in many of the key events there. The major naval bases further away from Petrograd were in the Black Sea, Tallinn and Helsinki, and while local centres of discontent, they played a relatively small role in the overall revolution.
Not so Kronstadt. The fortress, with a population of roughly 82,000 (20,000 soldiers, 12,000 sailors, 50,000 civilians), was a noted centre of discontent even before 1917. There had been a severe mutiny in the 1905 Revolution, which was somewhat unusual for the armed forces. The strict discipline of the Imperial Russian Army and Navy caused severe discontent in the ranks, and the economic and food situation had only made things worse. As the Baltic was the playground of the vastly superior German fleet, the Russian Baltic fleet was confined to mostly defensive work with the occasional convoy raid.
The February Revolution reached Kronstadt a few days after the major events in Petrograd had begun. Wary of a potential mutiny, the soldiers and sailors had been confined to their barracks, and machine guns had been set up at key points on the island. However, this had little effect. The workers, soldiers and sailors assembled in the main square in the early hours of March 1 and declared that the Kronstadt Committee was now in charge. A number of officers who tried to stop this were killed by their men, as were several policemen in an exchange of gunfire. The commander, Admiral Viren, was executed by firing squad, as was Vice-Admiral Butakov. One senior officer was thrown overboard to die in the freezing Baltic.
The Committee issued a number of orders to re-assert discipline, and sent a message to Petrograd indicating their support for the revolution. The Provisional Government appointed a governor in response, but he arrived to late to be able to establish any real influence in the face of the already established Soviets. The workers Soviet had already set a minimum wage and endorsed a policy of 'defencism' (ie only fighting in order to defend the revolution, not in order to make land or financial gains). The Military Soviet soon successfully asserted its own control over discipline and military matters on the island, shutting out the commissar sent from the mainland. They also passed a motion refusing to allow the unapproved transfer of units away from Kronstadt, thus securing their own power.
Over the months towards July, there were various political manoueverings on the island. The Bolsheviks increased their levels of support amongst the population, and greatly increased the number of deputies in the Soviets. During the April Crisis, over a note sent to the allied governments by Pavel Milyukov affirming Russia's intent to honour their annexationist treaties, the Kronstadt Soviet passed a motion strongly criticising the Provisional Government. In late May, they clashed with the Provisional Government and Petrograd Soviet over the extent of their independence and freedom to pass their own laws. Kronstadt was consistently more radical than the population of Petrograd, and was generally viewed as a thorn in the side of both the Petrograd Soviet and Provisional Government.
However, all this so far has only affected Kronstadt. In order to see the role of the Kronstadters in Petrograd politics, we have to move forward to July. The July Days were brought about by severe discontent over the poor economic situation, general chaos in Government, and the failure of a recent major offensive orchestrated by the Provisional Government. A number of army regiments and workers in Petrograd had come out on the streets in protest, and sent representatives to Kronstadt to urge their support. The more radical elements amongst them aimed to overthrow the Provisional Government and hand power to the Petrograd Soviet. However, they were unable to readily find a political leader, as none of the mainstream parties really wanted a full scale revolution at this point in time. Several thousand armed Kronstadters made their way to Petrograd, where they assembled in front of the Bolshevik headquarters. There they listened to a number of speeches by prominent Bolsheviks including Lenin, then made their way to the Tauride Palace, where the Central Committee of the Petrograd Soviet met. However, they were met by Viktor Chernov, a leading Socialist Revolutionary, who explained that the Soviet did not want to take power. Apparently, he was seized by a Kronstadter who exclaimed "take power when it's given to you, you son of a bitch", and was almost lynched, but for the intervention of Trotsky. However, they achieved little, and the main instigators of the march on Petrograd were arrested.
In October, however, the Kronstadt garrison played a truly decisive role. Over the course of the year, the Bolshevik party had steadily gained more support from the Soldiers, Sailors and Workers who were increasingly disillusioned both with the Provisional Government and the main socialist parties. When the time came for the Bolsheviks to launch their bid to take over in late October, the Kronstadt garrison and the Baltic Fleet in general was one of their main power bases. Kronstadt had more or less declared Soviet Power a number of weeks before October, and so it was only natural that they lend their support to a similar aim on the mainland. Kronstadt soldiers and sailors took part in the final assault on the Winter Palace and were instrumental in securing key areas of Petrograd, especially given the somewhat lukewarm attitude of the Petrograd Garrison itself to the Bolsheviks. Elements of the Kronstadt forces also took part in a number of skirmishes outside the city against troops loyal to the regime. A number of political prisoners taken during the October Revolution were confined in Kronstadt.
Kronstadt's third contribution towards the Revolution was in 1921, in the dying stages of the Civil War. While they had initially enthusiastically welcomed the Bolshevik seizure of power, and the transfer of all power to the Soviets, the Kronstadters became aware that Bolshevik power wasn't all it was cracked up to be. The arrest of other Socialists and increasing use of terror, combined with the critical food situation led to the Kronstadters becoming disillusioned with the Bolsheviks. In late February 1921, after a factfinding delegation to Petrograd had reported widespread starvation, repression and discontent, the crews of the Battleships Petropavlovsk and Sevastopol passed a number of motions calling for new elections to the Soviets and the lifting of various Bolshevik restrictions. These motions were passed by the wider island Soviets, who arrested leading Bolsheviks and seized the rest of the island.
The Bolsheviks acted quickly, seizing Kronstadt's main supply depot and closing off all roads across the Winter ice to stop any Kronstadters from escaping with news of their revolt. They branded the rebellion a counter-revolutionary one, and emphasised the number of Tsarist officers in positions in Kronstadt. The sailors in Petrograd proper were locked in their barracks and threatened with machine guns, and the Bolsheviks lifted some of the restrictions on Petrograd to ease discontent. The Bolsheviks rallied their most loyal forces and attacked across the winter ice. Despite taking enormous casualties, they succeeded in taking Kronstadt, and brutally repressed any remaining resistance. 9000 refugees walked across the ice to the now independent Finland, and safety.
We can see the major role that the soldiers and sailors had in events in Petrograd across the revolution and civil war. In the July days, they played a major role in protests and near-overthrow of the Government. In October their forces proved decisive in the fight for control of Petrograd, and in 1921 they were a major threat to an overstretched Bolshevik party at the limits of their strength. It's important to note, however, that the Russian Navy as an entity practically ceased to exist after February 1917 due to the breakdown in command. After February, the Russian fleet would take very little action, and discipline amongst the sailors was reported to be low. As such, while members of the Navy took a major part in the revolution, the Navy itself as an organisation did not.
Sources:
Paul Avrich, Kronstadt 1921 (2014)
Israel Getzler, Kronstadt 1917-21: the Fate of a Soviet Democracy (2002)
Christopher Read, From Tsar to Soviets (1996)