They would just dredge it like they do now. Sand bars and whatnot are constantly stacking up in the river.
The Corps can't keep up right now. Their dredging budget is constantly getting cut.
But yes, they'd just have to dredge more. Hell, there's not really any water in the river right now. The damn river is so low right now it's ridiculous. I think it was at 4.5 (feet above sea level) in Baton Rouge this morning. It's pretty much standing still, then when the tide goes out (>200 miles away) it lets a little water out. When the movement is only 150,000 cf/second, that's pretty freaking low considering how many square feet of area within the cross-section of the river in Baton Rouge.