Using +4V output, to trigger an npn transistor operating on +12V
An NPN transistor can do a great job of translating the 0-4V output of the sensor to 0-12V swing. See the circuit below:



Note that when your sensor activates and goes low on the output the output of this circuit will pull to +12V through the R3 resistor. 

Component values shown in the circuit are a nominal starting place for you to work with. You may have to adjust the resistors R1 &amp; R2 some based upon what the sensor output swing is once it has this circuit connected. R3 may require adjustment in the case the load you are driving is low impedance.