It's not too wide: the Netherlands actually did a study on this.
It is a pretty extreme proposal of course, but it is possible. It would have some big benefits, not just to Netherlands, but all the countries around it, because all of them (UK, France, Germany, Belgium) stand to lose a lot of land to rising sea levels. Instead, with this dam, they could actually create a LOT more new land in what's now the North Sea. Remember, much of the North Sea used to be dry land (we call it "Doggerland" now), only about 8000 years ago, and had human settlements.
I think shipping could still be an issue. Sure, dams can have locks, but the sheer amount of ship traffic there might be difficult. Maybe they should build new some ports and use freight rail to move cargo around.
But the cost would be extreme. Shipping access isn't a major issue though. A dam can have locks.