I can only speak for USNA but would guess their approach would be similar to USAFA.
If you need to work to help support your family, it is understood that you may have more limited ECAs and sports. You need to ensure that USAFA understands this -- via your ALO interview, your essay, etc.
However, if you "need" a job because you want additional spending money, I would probably go with the sport, assuming you must make a choice.
A couple of years ago, I worked with a candidate with an analogous situation. His only ECA was a particular sport. When I asked why he didn't do anything else, he replied that his father was a geographic bachelor and his mother worked long hours. So, after she went to work, he got his 3 siblings up, made them breakfast and then took them to 2 different schools before going to his own school. After school, he picked them up, helped them with their homework and fixed them dinner. He found time for the sport but not time for much else. He received an appointment.