Praise be to Allaah.
Yes, you could become Muslim and still play basketball, so long as there is nothing forbidden or harmful involved. Islam does not forbid sports that exercise the body and strengthen its muscles. Indeed, it is the body’s right that its owner should take care of it, as the Prophet of Islam Muhammad (peace and blessings of Allaah be upon him) said: "Your body has a right over you."(Reported by al-Bukhaari, 4800). One of the benefits of basketball is that it is based on aiming the ball correctly. So there is nothing wrong with playing it, so long as it is not combined with something forbidden such as gambling or deliberately harming or hitting others, and as long as it does not distract a person from praying and so on. By the way, I am sure that you are not unaware that there are a number of internationally known basketball players who have embraced Islam. Our advice to you is: don’t hesitate to enter Islam, for there is nothing to stop you. We ask Allaah to help you to follow the truth and to guide you to the way of goodness, for He guides whom He wills to the Straight Path.