Even if you're not doing the arch, because you're ...

Even if you're not doing the arch, because you're afraid of heights or claustrophobic, I'd recommend the museum.

I was travelling to St Louis on business and stayed at the Hyatt right next the arch. I found time in the morning to visit the museum and ride to the top of the arch.

The museum is very nice for a free museum, telling the history of that area from the time of native peoples to the current period. There was also a nice exhibit on the design and building of the arch.