Check to make sure there are no prospective boyfriends, elderly neighbors or Brownie troops with a line of sight to the proceedings. Though of course they're probably going to show up unexpectedly ANYWAY, once you're in the middle of things. Prepare a good explanation.
Trim your fingernails short. Assemble horse, hose, and your sense of humor (plus, ideally, Excalibur cleanser and perhaps thin rubber gloves).
Use hose (or damp sponge) to get the sheath and its inhabitant wet. Uh, that is, do this in a *civilized* fashion with due warning to the horse; he is apt to take offense if an icy-cold hose blasts unexpectedly into his personal regions.
Now introduce your horse to Mr Hand. What I find safest is to stand facing the horse's head, with my shoulder and hip snugly against the horse's thigh and hip so that if he makes any suspicious move such as raising his leg, I can feel it right away and am in any case pressed so close that all he can do is shove, not really kick.
The horse should be held by an assistant or by your free hand, NOT tied fast to a post or to crossties. He may shift around a good bit if he's not happy with Mr Hand's antics, but don't be put off by that; as long as you are patient and gradual, and stick close to his side, he'll get over it.
Remember that it would be most unladylike of you to simply make a direct grab for your horse's Part. Give the horse a clue about what's on the program. Rest your hand against his belly, and then slide it back til you are entering The Home of the Actual Private Part. When you reach this first region of your destination, lube him up good with Excalibur or whatever you're using.
If the outer part of his sheath is really grungy you will feel little clods and nubblies of smegma peeling off as you grope around in there. Patiently and gently expedite their removal.
Thus far, you have probably only been in the outer part of the sheath. The Part Itself, you'll have noticed, is strangely absent. That's because it has retired shyly to its inner chambers. Roll up them thar sleeves and follow in after it.
As you and Mr Hand wend your way deeper into the sheath, you will encounter what feels like a small portal that opens up into a chamber beyond. Being attentive to your horse's reaction, invite yourself in.
You are now in the inner sanctum of The Actual Private Part. It's hiding in there towards the back, trying to pretend it isn't there. Say hi and wave to it. No, really, work your finger back and forth around the sides of it. If the horse won't drop, this is your only shot at removing whatever dried smegma is clinging to the surface of the Part itself. So, gently explore around it, pulling out whatever crusty topsoil you find there. Use more water and more Excalibur if necessary to loosen attached gunk.
When Mr Hand and the Actual Private Part have gotten to know each other pretty well, and the Part feels squeaky clean all around, there remains only one task: checking for, and removing, the bean. The bean is a pale, kidney-shaped accumulation of smegma in a small pouch just inside the urethra. Not all horses accumulate a bean, but IME the majority do, even if they have no visible external smegma.
The equine urethra is fairly large diameter, and indeed will permit you to very gently insinuate one of your slimmer fingers inside the urethral opening. Do so, and explore upwards for what will feel like a lump or "pea" buried no more than, I dunno, perhaps 3/4" in from the opening.
If you do encounter a bean, gently and sympathetically persuade it out with your finger. This may require a little patience from BOTH Mr Hand AND the horse, but the horse will be happier and healthier once it's accomplished.
Now all that's left to do is make a graceful exit and rinse the area very thoroughly in apology for the liberties you've taken.
A hose will be MUCH easier to use here than just a sponge and bucket, IME. Make sure to direct the water into the Part's inner retreat too, not merely the outer part of the sheath. This may require you to enfold the end of the hose in your hand and guide it up there personally.
Ta-da, you are done! Say, "Good horsie" and feed him lots of carrots. Watch him make funny faces at the way your hands smell. Hmm. Well, perhaps there is ONE more step...
The only thing I know of that is at all effective in removing the lovely fragrance of smegma from your hands (fingernails arms elbows and wherever else it's gotten) is Excalibur. Even then, if you didn't use gloves you may find you've got an unusual personal perfume for a while.
So, word to the wise, do NOT clean your horse's sheath just before an important job interview or first date.
Of course, there is that one FINAL step...
Figure out how to explain all this to your mother (or the kid from next door, or the meter reader, or whoever else you've just realized has been standing in the barn doorway speechlessly watching the entire process.
Now, go thou forth and clean that Part!