Long time STL guy here. There is nothing family friendly in downtown at all. Forest park is ok and the zoo igreat but make sure you are out of there before dark. North and east St. Louis are worse than detroit. do not go to either. The science museum is loved by kids and adults alike. If you are into art of history those museums are quite good too. Oh, they are free also. Of course the mighest profile tourist attraction here is the arch. It is worth dropping by and checking out the museum underneath it. If you have no hint of claustrophobia then take the tram to the top of the arch. Sure it is a heck of a view and quite interesting but the most memorable part of the experience will be the ride up (heh heh heh, just ask anyone who has done it)
Since you will be at Barnes hospital you should know forest park is right across kingshighway. Unfortunately the warnings about forest park go for the area around the hospital.
A few random statements about the area.
DO NOT go more than a couple blocks north of the hospital, bad mojo there.
If any area you wande into seems bad it probably is.
The layout of the roads in the area make no sense at all. Get a local map and carry it with you whenever you drive anywhere.
It is hard to find good places to park downtown.
There are great places to eat in StL, check out some of the finest italian cuisine in "The Hill" area. (Oh yeah, you can wander around there after dark)
The Landing area is hit or miss but you with kids I see no reason for you to even bother to check it out.
Like to gamble? Loads of riverboat casinos all over.
The Zoo is great, so is the Arch.
Catholic? check out the Basilica.
Grants farm is great for animal lovers.
The tour at the A_B brewery would be a wet dream for most of the beer swilling alkies here. (Heck, it is nice even for more casual drinkers too)
StL for some unfathomable reason has done nothing at all with its riverfront. Nothing down there but warehouses and winos.
STL has a lot of gunshows, see if there is one around when you are here.
Keep away from St Louis Center, union station is ok.
Artistic? Laumeier sculpture park is interesting to those so inclined.
Downtown St. Charles and the antique area of Alton, Il. are really quaint for lovers of overpriced antique type junk. (seriously, I don't get it but antique lovers just gush about those places)
Hopefully this will help, any more questions feel free to ask.
ETA;
The city museum is privately owned and charges a fee though I am sure it is not very much. I have heard it is interesting but it is also full of rambunctious kiddies so if you enjot the screaming of a thousand ankle-biting hooligans then you will love it.
Gary(the obviously childless)M