Casino Queen worked really well for us. You can see the Arch across the river from the campground.
They'll shuttle you, very short trip, to the Metro station and after riding the train a few minutes across the river, you'll get off at the first stop and walk to the Arch through a park-like area. (Call when you return for a shuttle back to the RV campground.) If you want to drive over, there are some parking lots available.
The museum under the arch is very good with interesting films, though temporarily closed for renovations according to the website. Lots to learn there about Jefferson, the westward expansion, and Lewis & Clark, and also about the engineering and construction of the Arch.
Definitely ride the tram to the top!
Another fun thing to do - take a short riverboat ride on the Mississippi. Board just down the bank from the Arch. There's a little sandwich place there while you wait.
