Although Basecamp doesn't officially support subprojects, there's an easy way to create them. Use companies as categories, and the projects assigned to these companies become subprojects for that category.
For example, let's say that your company, Mango Design, works on both web design and print design projects. You also want to use Basecamp to manage several internal projects. You could create companies called "Web Design Projects" and "Print Design Projects" to act as categories for your subprojects and use your main company to house your internal projects.