what sort of switch can I use?
Arduino pins set as INPUT does not eat much current, when you do not read them. 

If you are not to go for power-save sleep modes and similar technics, you are probably good just to connect one resistor between  3.7V and analog pin, then other between analog pin and digital (or analog) pin. Set the other pin as INPUT and it would be high resistance and nearly no current will go. 

When you want to read voltage, set the digital pin to OUTPUT and LOW and you have ground. Wait for a moment to things to set (I do not know, like delay(5) or so), read the analog pin and sent the digital to INPUT again (and you do not consume power again).

Other solutions may be much harder, need other circuits (like relay) and even if such circuits  would not eat power in disconected state, the cost (in power spend) of switching them may be higher, than the minimal power consumed this way. (Especially when you need a pin for managing the switch anyway).