In our area we have a lot of youth only programs: gymnastics, dance, karate, yoga…. and so on. Many of the exercise studios offer kid friendly programs, for example I know of a pilates place that offers teen specialized classes, and a number of yoga places that do. For general workout places families with kids go almost exclusively to family styled gyms. Remember that a big reason the Y is so popular is that the parents can come to the same place with the youth and do their thing with the babies in the baby room, the toddlers in the pre k classes, the youth in youth classes, or in the lobby doing homework, and the older youth in the gym.
And in families where the parents do not work out it is either because of lack or resources or lack of interest. In either case the kids are usually not motivated to ask to join a gym, though they might ask to dance or do gymnastics because their friends do. To get people in your door you either need to appeal to the parents, or to the kids, or to have a school or corporate connection that will help monitize what you offer as well as advertise it.
Are you thinking of a kind of upscale ‘botique’ place where people drop off kids? You will need to think about staffing accordingly.
Are you thinking of a more family friendly place? Here you get back to the question of money and motivation. One big question is do you have Ys and/or JCCs in your area? They tend to have lots of kids programs, and lots of resources to promote them. And due to quite a lot of factors provide their programs usually very inexpensively. Our local Y has a large teen program and a ton of the town teens (of all ages) use the facility.
It could work really well, depending on your area, how carefully you do your research, and so on … but I think as Karen says you need a careful business plan, and especially to look into what is available and what people want.