I'm sure that both the civilian and AF bases have ILS systems. It's entirely possible -- though entirely ridiculous -- that the pilots simply looked up the wrong ILS frequencies on their sectional chart during their pre-flight planning. It seems the ILS for KRCA (Ellsworth) Runway 31 operates on 110.30 MHz, while the ILS for KRAP (Rapid City) Runway 32 operates on 109.30 MHz. The frequencies differ only by a single digit, so a simple entry error might be to blame. Rapid City's Runway 14 (the same runway as 32, just in the other direction) does not have an ILS approach.
Commercial aircraft do indeed fly to decision height automatically on a Class 3 approach. If they were on the ILS in clouds, it is possible that they just never realized they were on the wrong approach -- and once you're close enough to make visual with a runway, it's pretty hard to tell what airport you're at -- runways all look pretty much the same. These two runways, however, are not numbered the same -- the regional airport's runway is 14/32, and the AFB's runway is 13/31. The pilots should have noticed this when they made visual and done a missed approach, but, well, frankly... you land at so many airports so many times, you don't really think to check the number on the runway after you've completed a clean ILS approach!
The tower is not in continuous contact with the pilots during a landing -- they clear the aircraft for an ILS approach, but that's about the end of their involvement. Rapid City's 32 ILS does have an outer marker, called "Ranch," that most pilots would report crossing. They may not have worried about crossing the outer marker, or they may have thought it was malfunctioning. Since the two airports are so close, it's actually conceivable that they may have crossed KRAP's outer marker as expected even while on the localizer for KRCA. I'd have to have approach plates for the area to verify this, but I don't have them (I don't live in SD!).
You'd have hoped that the pilots would have noticed all of the other information that their plane was giving them to indicate they chose the wrong ILS -- their heading would have been wrong, their GPS should have giving them a large course deviation indication, etc. They may not have been using those tools properly, however, or at all. Keep in mind that virtually all aviation mistakes are 100% pilot error.
While there doesn't seem to be an NTSB report on the mishap yet -- and perhaps there won't ever be one, seeing as it didn't really qualify as an accident -- I would bet money that the pilots just entered the wrong ILS frequencies into their instruments.
- Warren