Odds of Getting In
Breaking in to the dredging business isn't a tall order. No degree necessary, no experience necessary. What is necessary is a job opening somewhere, and the hardest part about getting in is probably the lack of folks who are getting out. People who land a job as a dredge operator and don't wash out right away tend to stick around for decades.
So how does it look on the new-jobs-front? According to the U.S. Department of Labor, job openings should grow by about forty new dredge operator jobs per year between 2012 and 2022 (source).
Those are pretty stark numbers, so if you're not able to land a gig with a private company or the U.S. Army Corps of Engineers, you can gain valuable experience in the construction business. Anything that allows you to work with heavy machinery will give you an edge over other applicants when a dredge operator position does open up somewhere.
Oh, and it'll help if you like to travel. You might find a local job, but a more likely route is you'll go to where your dredge is called. That could be anywhere in the country, and in some cases, the world.