Connecting thermostat common wire to air handler or boiler?
From your picture it looks like the C from the air handler is simply connected to the boiler C. So you could pickup a C for the thermostat at either the air handler or the boiler. 

The main problem you are going to have is related to turning off the electrical supply to the air handler. You will lose the 24VAC power feed to the thermostat if you do that. So instead you want to leave the air handler powered but program the new thermostat so that it does not turn on the demand connection to the air handler when you do not require its usage. This should result in very little standby electrical power consumption but allow your new thermostat to work. 

The "demand connection" I refer to is the thermostat output that is turned on to request heat, cool or in some instances fan. You may need to check if the air handler has the fan running when you think the thermostat has both the heat and cool modes shut off.

Another thing you should check is if the thermostat will run with just one of the Rh and Rc wires connected. Check to see with the air handler power shut off if the Rx wire from the boiler still has 24VAC.

I see that you have the G wire connected to the air handler. As I recall the G wire is for running the fan. You need to see if the air handler stops blowing out air if you disconnect the G wire when the AC power is turned on to the handler. It could very well be that the thermostat activates the G wire in heat mode despite your boiler heat source not using an air circulation fan. A truly smart thermostat may have a programming option to not drive the fan in heat mode to support boiler applications.