well is it compulsary to use an arduino ? other wise the reedswitch thingy can be done without an arduino .
or a motion sensor.
a motoin sensor detects movement i guess you dont want to display that the door is open when someone walks past
anyways using an arduino or not depends on what you want to do gove us more details if you want something like a alarm when the door is open or a led to light up then you can do it without an arduino
The local DollarTree store often has alarm and led light gizmos similar to the below for $1. I think the magnet in these holds the reed switch open when the door is closed.