Here's the way I prove that the fuel pump is the problem (listed in to particular order):
1. Swap the fuel pump relay with another one in the power distribution box. Also examine the relay's socket for corrosion.
2. When the vehicle wont start, try spraying starting fluid into the throttle body. If the the engine them stumbles to life, it's not getting fuel, which rules out an electrical problem (like the crank sensor).
3. Check the inertia switch's connections to make sure they're not loose, burnt or corroded.
4. When the engine wont start, try turning the key OFF/ON multiple times (may take up to 20+ tries or more) until you hear the fuel pump prime.
5. You can try testing the wire connection found on the frame rail behind the driver's side rear tire for power when the key is turned on. (requires two people and a VOM).
I've replaced the fuel pumps in all 6 if my Explorers/Mountaineers. 4 of them demonstrated the same symptom you're experiencing (1 flat out died and 1, with 260,000+ miles on it, turned out not to be the issue, but at that mileage was certainly due to be replaced). They all crapped out when the weather turned cold and all had between 160,000-200,000 miles on them.
BTW, I replaced the fuel pump on my '01 ST at 183,000 miles. Did it myself and my total cost was about $80, which included a quality fuel pump, a new strainer sock and a hard to find 3-port fuel filter.