Aside from a visual inspection of the shocks (and other suspension components).
What I do is push the car down on the fender with my hands and body weight and see how it comes back up.
But that check can really only tell you anything when shocks are AWFULLY bad.
If you feel the car handles all fine and good, especially when going over bumps or when breaking hard, then the shocks should still be alright.
And if the car still handles really well when you're driving it quick on a bad uneven road or even on a dirt road, then you can be sure the schocks must be fine.
Here are two nice videos (pt1 and pt2) by the way that might be able to help give you a little bit of an impression, if possible at all in a video:
www.youtube.com/watch?v=vVVgoB-BjXg
www.youtube.com/watch?v=Vx4UwV8Ex7s
(The video also continues with pt3 and 4 by the way.)
But then again, this video is not of an SUV or truck like the Explorer.
Naturally an SUV will not handle quite like a sedan, being heavier the center of gravity higher up and the suspension also set up with off road capabilities in mind and not just street handling. But if you can at all find helpful answers to your question just online, then this video in it's 2+ parts is probably a pretty good resource on what to look for.
Another general thing, when you check how the car handles with then intent of figuring out if your shocks are fine: Keep in mind that other components of the suspension of course also have their functions and effects on the vehicles handling.
For example the roll of a car will be affected negatively by bad shocks, but of course it could also (and even more so) be affected if old sway bar links failed.
And whenever you "check" the suspension while driving, please be critical and really expect a car to handle like it was all "new" and not just "somewhat okay" for a car that old.