You can't. Few (if any) OSes provide that kind of granularity. Those decisions are typically left to the discretion of the network layer, which makes no distinctions between apps (it's not even aware of which apps are actively using the network).
The only real way to use different network connections for different browsers (or any other kind of apps) is to use different machines. So what you could do is use virtualization (e.g., VirtualBox). Since virtualization lets you control which network adapters are passed to the VM (virtual machine), you simply determine which network adapter has priority in the host machine, and pass the other network adapter (w/ the lower priority) to the guest machine. Voila, both the host and guest are using different network connections. In fact, using VirtualBox and seamless mode, you can make it appear as if apps from the guest environment are running on the host.