Synonyms are not always freely interchangeable. Sometimes there may be different shades of meaning. Sometimes different grammatical forms may be needed.
In the case given in the question, however, "besides" and "aside from" have essentially the same meaning, and can be substituted without changing the meaning of the sentence:
Do you play any other sports besides basketball?
Do you play any other sports aside from basketball?
Also, the word "other" is redundant here, and the following also have the same meaning:
Do you play any sports besides basketball?
Do you play any sports aside from basketball?
All of these more or less imply that the person addressed (that is, the person who is being asked the question) does play basketball, although not absolutely. The following example makes that more explicit, but otherwise has much the same meaning:
Do you play any sports in addition to basketball?
Which of these alternatives to choose is a matter of style, not of grammar or meaning.