This is a complex topic - the RN's efforts to combat slavery extended over most of a century, and vast areas of sea - but it's possible to summarise by saying that the efforts made were serious, but that they were only truly effective when placed in tandem with diplomatic initiatives.

What do I mean by "serious"? I don't think it need be doubted that many RN officers were genuinely opposed to the slave trade – not least those who actually witnessed its effects up close – and so were zealous in their efforts to stamp it out. Equally important, however, was the fact that, in the century of the Pax Britannica that followed the end of the Napoleonic Wars in 1815, opportunities for promotion in the RN slowed radically, a product both of the extraordinary expansion in the number of ships and officers that had taken place during the war period, and hence of oversupply of men qualified for the vastly reduced number of posts on offer in the postwar navy, and of lack of opportunity to distinguish oneself.

For much of this period, performing acts of bravery and daring on antislavery patrols was one of the few ways that a junior officer could draw attention to his merits, and thus hope to secure early promotion and leapfrog his contemporaries - hence securing better pay and better opportunities faster than would have been the case if he moved up the ranks purely as a result of seniority. It helped that the sort of ships used for anti-slavery patrols were small, fast units of the sort given to lieutenants and commanders, rather than captains, to command. In addition, securing prize money – as happened when one captured an enemy ship – remained the quickest way of making significant extra cash while in service. So RN officers had private and professional, as well as moral and ethical, motives for taking anti-slavery patrols seriously.

All that said, the sheer geographical scope of the slave trade, and the active efforts of local rulers and other western nations to continue to profit from it, combined to reduce the effectiveness of RN efforts. The Transatlantic Slave Trade Database notes that

the traffic could be halted only by the intervention of the governments of regions that were either exporting or importing slaves; it could not be halted by naval action alone

and adds the datum that

naval intervention did result in the capture of nearly 2,000 slave vessels after 1808. Only 544 of these had slaves on board at the time of capture, but their 125,000 captives (or strictly, re-captives) were diverted from the sugar and coffee plantations for which they were intended, and for the most part ended their lives with choices they did not have prior to their re-capture.

The same source notes that "between the 1840s and 1850s, the traffic declined from an average of 50,000 a year to 16,000, and after 1860, to half this," but points out that its decline had more to do with diplomatic pressure placed on other powers to limit and then halt the trade:

Beginning in 1810, the British established a network of treaties that allowed their naval vessels to detain the slave ships of other nations. The decisive actions against the traffic nevertheless did not come until the mid 1840s and again in 1851, when the Cuban and Brazilian governments respectively took serious action against the slave trade.

Sources

John McAleer and Christer Petley, The Royal Navy and the British Atlantic World, c. 1750-1820 (2016)

W.E.F. Ward, The Royal Navy and the Slavers (1969)