For how long? A season, a week, a night?
Assuming a short term stay, your best bet is Barkers Marina. They welcome transients. Its on the Superior side. Full service joint with store, showers, restaurant, and bar. In Duluth proper, Harbor Cove and Lakehead often do not have staff there every day to receive you. Lakehead DOES have a long dock you might be able to tie to.
Up the St Louis river, Spirit Lake Marina will often have slips, but its an hour motor.
Up the coast is McQuade harbor, run by the state. Not sure if you can spend the night.
We do this all the time; its actually rather HARD to moor in Duluth and not have a $50 taxi ride into town. We time our visits so we can use the slip of a friend in the Duluth Slip Harbor, a private marina right downtown.