Hole In Wall Under Bathroom Sink
A hole in the wall underneath the bathroom sink, like the
picture above, doesn't create a structural problem and it really doesn't
create a cosmetic problem, as long as the cabinet doors remain closed.
It does however create a problem for insects and rodents. If this area isn't repaired and sealed up correctly, there's a small chance, that this could be the perfect home for a rat or mouse.
If you don't have a rodents or insect problem and it doesn't bother you to look at a hole in the wall under your bathroom sink and this isn't up against an exterior wall, it's your choice, whether you think you need to fix this or not.
If the other side of the bathroom wall is the building exterior, you could be creating a problem with your home insulation. This large hole underneath the bathroom sink could provide a nice spot for exterior air to enter your home. This could eventually create problems with your heating and cooling bills.
Home Repair Tip: If you just wanted a quick fix that wouldn't cost much money and you didn't really care what it looked like. You could always go down to your local home improvement center and purchase some insulation foam to spray into the hole. The foam will expand and seal up the hole.
Another home repair method would be to, cut a piece of drywall, install it and finish it, to make the correct home repair.
Another option would be to glue, screw or nail a couple of scrap pieces over the hole, to cover it up. Be careful though, if you're using screws or nails, to avoid hitting any of the plumbing pipes. Sometimes glue might work better in situations like this.