I did this a few times. (although SD was never the destination, I was always passing through going farther west or on the way back.)
As was already said, the drive is extremely boring, although for kids under 5 that shouldn't matter too much. I would advise you to try and break it up with a stop in/near ORD -unless you're cool with driving 30 hours straight.
I don't know how much time you have or what kind of schedule you're planning, but you should consider leaving NY Wed night/Thurs and spending Shabbos somewhere around ORD then continuing on after shabbos. Basically once you pass ORD there's absolutely nothing to see or do until you hit the badlands area.
A few ideas for activities off the top of my head:Mt Rushmore itself is
at most a half-day activity.
There's this really nice fairy tale themed park in Rapid City (free admission IIRC)
There's also a nice little dinosaur park around there somewhere. I remember it being up on a hill, I think right outside Rapid City.
I don't know if your kids are too young for these.
There's a ton of tourist stuff to do around the Keystone area but I was always too cheap to pay for any of it
I don't know if this is your type of thing, but there's a really nice Air and Space Museum just outside Ellsworth Air Force Base. I'm pretty sure it's free admission otherwise I probably wouldn't have gone multiple times
One thing they offer is a (paid) guided tour of a minuteman missile silo on the Air Force Base, I really enjoyed that.
(side note, One time when I was there, they were doing some training or something, they were circling around then touching down on the runway but aborting and taking right back off. This was going on for quite a while when I was outside the museum watching. I found it super-cool.)
There's also the Minuteman Missile National Historic Site around there, unfortunately it never quite worked out for me to make it there yet...
And of course after seeing seventeen thousand billboards you just have to stop at Wall Drug

nothing spectacular there but some things are kinda cute/interesting.
One time at Custer State Park I saw tons of bison, the next time zero. (Or maybe that was Wind Cave NP, I'm not 100% sure)
Of course there are the cave tours there. IIRC the park admission is free but the caves cost money.
If you go down to Wind Cave and pass through Pringle, stop at the Pringle General Store, you'll probably see the same guys sitting around playing poker from when I was there.

(and they have a chair that's supposedly the one Wild bill Hickock was sitting in when he was shot)
Gosh I
really need to do a road trip, it's been way to long
