I don't often find myself asking a question that would elicit this specific answer, but in casual conversation, I might simply ask "Do you have any brothers or sisters?" and after their answer, as a follow-up question, you might ask "Are you the oldest or youngest?" Speaking as an American, culturally, birth-order is not very important (only important in an arbitrary manner; depending on the person), so you would probably not have that sort of conversation very often. You could potentially ask very directly "In what order were you born, in comparison with your siblings?" But that sort of question would probably be seen as very odd and unnatural. In any case, simply ask about their siblings and that information may come up, but not always.

"do you have any brothers or sisters?" Then the person will say yes or no or yes I have X (sisters/brothers) I have X (younger/older) (sisters/brothers) Then you can work it out or you will have to specifically ask