Is there any way to permanently remove otter from a watershed?

No.

put up a wall as Trump has suggested.

Remove all fish, shellfish, frogs, etc and you won't have any otter. But that sounds like a terrible idea.

Just trap them out as they show up.
Multiple sets at strategic funnel points will take all the otter using a small watershed at any one time.In cases like that trappers must be extremely careful not to overharvest.This is even more critical if trapping in winter after freeze up as Otters travel routes under the ice are even more restricted,and sets such as the Mallette ice tunnel will take every otter going up and down stream on a medium to small watershed.
Otters travel between watersheds so there will always be some returning after you wipe out the ones using the watershed as long as the feed source is there.
On big rivers it would be impossible to harvest all the otters using such a large watershed.

The reason I asked this was because I was checking in with my landowners to make sure I was good to trap again this year. A guy that has an otter problem said he’s looking for a way to exterminate the otter and if I had any idea to let him know. Told him only way I knew how was to trap or shoot them, but that being a water shed he’s always going to have some form of problem with otter

Build a catfish pond nearby.