According to Wikipedia, from St. Paul to Baton Rogue, the Corps of Engineers maintains a depth of at least 9 feet (since the 1930s). That said, it's much deeper in most places. For example, the average depth is about 120 feet. In St. Louis, the average is about 30. Even as far north as Minneapolis, it's 16 feet. It gets somewhat shallower as you get closer to New Orleans. Throughout its course, you get sand bars and other obstacles that shallow the river.
I know that doesn't help much if you're interested in the 1800s but my guess is that you'd be safe for much of the river with a draught of about 10 feet. Before the 1850s though, you'll not be able to get anything north of Minneapolis.