There were a number of reasons:
The Russian annexation of Constantinople and control of the strait between the Black Sea and the Mediterranean was a long held ambition of Imperial Russia (although not all Tsars pursued it) and constant fear of the British during the 1800s. Britain and Imperial Russia were colonial rivals in Central Asia (known as the "Great Game") and it was feared that, if war were to break out, Russian control of the Straits would allow Russian fleets to cut British communications with India and seriously weaken the Empire. Therefore control of the Straits could not be allowed to fall into Russian hands.
There was the ongoing issue of the " Eastern Question" - By the mid 1800s it was clear that the Ottoman Empire was a spent force. It was weak, technologically backward, in economic difficulties and full of rebellious minorities. But it was also clear that the break up of the Empire would risk war between the Great Powers. Therefore there was incentive for the Powers that did not have territorial ambitions in Near East to try to prevent the Ottoman collapse. Britain, and later Germany (with the Treaty of Berlin), worked to shore up the Ottoman Empire, and thereby prevent a war they would likely get drawn into and gain nothing from. A Russian occupation of Constantinople might trigger the collapse of the Empire, and therefore the British sought to prevent it.
In 1853 the British (and French) had gone to war for the reasons outlined above (the Crimean War). The war ended with the Treaty of Paris, which banned the Russians from launching ships or maintaining a fleet on the Black Sea. But, following the French defeat in the Franco-Prussian War, the Russians announced they no longer considered themselves bound by terms of the Treaty of Paris, and would rebuild the Black Sea Fleet. This, coupled with issue 1, was seen as a threat to British naval dominance of the Mediterranean, and therefore preventing Russian control of Constantinople was more important than it may have been in the 1860s.