Connections has not been tested with WordPress Multisite, but is has been reported by several users that Connections does work without issue. Please follow this process to properly activate Connections on WordPress Multisite.
If you have activated Connections for the network, you have to disable the network activation. Then go into the multisite settings and check the box for plugin administration. Next go into individual sites you wish to use Connections, you can then choose which non-network activated plugins you want to activate. Connections should show up in this list, activate it.