Click the Tunnelblick icon in the menu/status bar, then click "VPN Details".
The "VPN Details" window will appear. Select the connection(s) you want this to be effective for in the "VPN Details" window, then click the "Advanced" button. (To select more than one connection at a time, hold down the "Command" key while you click.)
The "Advanced" window will appear. In its the "Computer sleep/wake" box, remove the checkmarks from "Disconnect when computer goes to sleep" and "Reconnect when computer wakes up".
Close the "Advanced" window by clicking the little red circle in the top left corner of the window, then do the same for the "VPN Details" window.