I'll start with a really minor one change: I'd tweak the escape from the circus train. I'm never quite sure if I'm not understanding it, or if it was not edited right, but why does Indy come running onto the tracks from the side? I assume he's supposed to have escaped through a trap door beneath the magic box, so he should be coming out from under the train. Just trim a second or two from the shot of Indy running away so that we don't see him running on from the side.
(Even then, the whole concept is a bit silly since why would the train have a trap door under the box? Presumably the box is unloaded from the train and used at the circus, so a trap door in the train would be completely pointless. But that's just nitpicking, and missing the spirit of this sort of movie.)
On a larger scale, I'd cut back on SOME of the humor throughout the film. Not all of it, of course, but enough to find a better balance so that it doesn't detract from the excitement. I think that was what disappointed me most about LC in 1989-- the tone of the movie was just TOO light-hearted and lacking in tension. I was hoping for something more similar in tone to Raiders or TOD (obviously there was a ton of humor in TOD, too, but it was offset by the darker elements and several scenes that really ratcheted up the tension in a way LC never did.)
Finally, as some other people mentioned, some scenes just felt surprisingly cheaply made at times. The prologue on the boat, for example, felt way too much like a soundstage. The motorcycle chase feels like what it is (i.e. something they added in quickly after the rest of the movie was done.) The biplane sequence effects were dodgy even for 1989. If they could get these up to the quality of some of the more impressive moments of the franchise, it would benefit the entire film.