It occurs to me that the more interesting question is not “Why do British people not like basketball?” but “How and why do some sports become popular in some places but not in others?”
Just to take a few examples of sports the originate in England:
Rugby was invented at the English public school of the same name, which under Thomas Arnold had become the model to be emulated by every other public school in the British Empire. Consequently, the game became popular among the upper echelons of British society and was exported to much of the British Empire. Curiously, however, rugby union, the form of the game played at the public schools and universities, also achieved universal popularity in Wales and the west of England, while a different form of the game, rugby league, became popular among working-class communities in the north of England. Rugby is very popular in South Africa, Australia, New Zealand, Tonga, Samoa, and Fiji, but it is curiously not as popular in Canada, the Caribbean, the Indian subcontinent, and Africa (outside of South Africa), despite these countries and regions having experienced similar cultural influences during the period of British rule. Rugby also has limited popularity in some other countries, e.g. in Europe and south America.
Cricket, on the other hand, is the most popular sport throughout the Indian subcontinent, again, due to the legacy of the British Empire. It is also popular in the Caribbean, which fields a strong side despite its small population. Why did cricket, but not rugby, become popular in the Indian subcontinent and the Caribbean? Again, cricket is not particularly popular in Canada, but it is popular in South Africa, Australia, and New Zealand. It is interesting that Canadians have relatively little interest in the two sports most closely associated with the cultural legacy of the British Empire.
Of course, the most popular sport in the world is association football. It is the most popular sport in almost all of Europe and Latin America, the whole of Africa, and much of Asia. Curiously, regions of the world where football is not the most popular sport include Canada, the United States, the Caribbean, the Indian subcontinent, and Australasia. How did football become the most popular sport in regions of the world where the British had little, if any, influence, such as Latin America and Russia?