Would replacing old mortar for new prevent water from seeping through brickwork?
It wouldn't hurt. looks like a lot of that mortar is just missing with holes and cracks. Looks like rain and a bit of wind could drive rain right through it. 

That said, it looks like the brick is also damages. Brick with a damaged face will also take in water like a sponge. They do make spray-on brick sealant. I can't say it it works or not, but might be something to look into.