I started on this subject some yesterday, mostly focusing on clothing. Sports is another area that can be expensive if you aren't willing to set limits. This seems to be more of an obsession of affluent parents, especially with sports because they feel it will make their children more "well-rounded" when they apply to a university. In moderation, that's fine.
Unless sports are a child's driving interest (and in that case, they usually are obsessed with only one or two), there's no reason not to exercise a little restraint here. One sport at a time should be enough for most children during the school year, though since most of them are seasonal, this can still allow for a child to participate in more than one over the course of a year. In the summer, it can be tempting to sign a child up for several things, but even then, one team sport and one individual one like swimming are probably plenty.
The idea here is quality over quantity. A child who is going to a different sports practice every night of the week is unlikely to be very good at any of them because he or she never focuses on that one sport. Ask yourself what the real goal is. Realistically, is your child a gifted athlete in that sport? If not, is the goal for the child to get physical exercise and learn to love a sport that he or she continues to play for life? Is it learning team-work? Is it filling out a resume? Is the child playing a sport for the love of it, or because you think he or she needs to?
Honestly, this is as much an area to set limits as clothing. If you just want a child to learn a sport in order to stay fit and have fun, don't jump into an expensive one. Some are VERY expensive. Some are quite cheap. Many don't actually require joining a team---swimming, skating, hiking. If you want to fill out a resume for college, you'll need to join a league sport, but you still don't have to choose an expensive one.
However, I'm not saying to refuse a more expensive one if a child really sincerely wants to participate in that sport and you can afford it. You can still limit it. If a child really wants to fence, for instance, see if you can rent equipment for a few months until it's clear he or she is going to stick with it. A child may try a sport to discover he or she doesn't like it, that's fine (but a good reason for a kid to play some backyard softball before signing up for a league). Once or twice is understandable. But I'd hesitate to spend a lot for a child to keep trying new sports if he or she never sticks with one beyond a couple of months. At that point, it's time to set a "sports" budget or quietly insist the child help pay for the equipment, lessons, fees, etc.
Honestly, unless there's a particular team sport a child wants to try, if all you want is for your child to be fit, why not buy bikes for both of you? You're probably spending that much time driving to practices and waiting for them to be over. This way you can both get fit, probably in less time, and have a good time together. If you want to use this to fill out a college resume, you can sign up for charity rides or bike clinics for young kids. Also, don't let a child's sports practice schedule overrun your life. If that starts happening, talk to other parents you're friends with to take turns carpooling the kids. And if the sport becomes stressful, it's time to reevaluate.
I do want to add that I have reservations about many organized team sports for young children; I question whether most young kids actually learn the lessons of teamwork and good sportsmanship that are claimed for these things. However I played several years of intramural basketball from the ages of 12 to 15 and loved it, and though I never considered swimming competitively, I spent every minute I could swimming for many years. I do have serious issues with the growing trend to "private" sports leagues that charge a high fee, effectively locking out poorer kids without having to openly discriminate as well (but potentially allowing them to offer scholarships to exceptional kids, thus siphoning off the best athletes from the open leagues), while often playing on tax-payer funded public sports fields.