I believe that you could use either phrase in answer. In this case, a or the would mean the same thing. Either implies that there is only one house with a red door (within sight). Your meaning would be completely understood.

I would be inclined to say 'the one with the red door' . There is just more emphasis when using 'the', and in this example, you are drawing attention to a particular house and its distinguishing feature. But really, it is a minor point - either is fine

Sorry, the title should be 'The one with a/the red door'.