Even for such a warm locale, the park's traffic increases dramatically in the summer. Like most destinations of its kind, the educational and scientific aspects -- as well as the breathtaking natural beauty, of course -- mean a lot of family vacations, as well as the sojourns of hikers, rafters, and outdoor types, will include stops at the Grand Canyon.
The majority of casual visitors, year round, tend to cluster around the South Rim's famous vistas and the West Canyon's tourist attractions, such as the Skywalk owned by the Hualapai Reservation. Families with children don't tend to take the trails too far down into the Canyon, of course, so you can expect the crowds to thin out fairly quickly on your descent, but the midsummer tourist season is alive and well up above the rim.
For those taking the trails down to the bottom, or even just half-way (a popular way to do it) into the canyon itself, the good news is that solitude reigns. Until you meet up with all the other hikers at the bottom, where the major trails all meet, your hiking party could very well spend the day's trek interrupted by only a few encounters.
If the park's emphasis on solitude is what appeals to you, just remember the guidelines for safe hours to hike during the day. Landmarks and waystations along the path include posted suggestions about the best times to make each segment of your journey, and park rangers are an amazing resource. Don't be shy about asking questions -- they know you're there to have a personal experience, and they have the know-how to tell you the best way to make it perfect.