What is the best way to wash my horse's face?
The best way is the safest way. This will vary from horse to horse. Some horses drop their heads and love a shower there. Others would rather kill themselves than get their face wet. One of my horses will only allow a face washing from one side. It's a total no-go on the other side.
Long before you ever think about using a hose around your horse's head, he needs to tolerate being handled and touched on the face. It's all about the baby steps. Then he needs to be OK with sponges and wash cloths on his face, eyes, nostrils, etc. Then you can dampen a sponge and wash cloth and try that. Often, you don't even need to go beyond using a sponge on his face. Remember, this handling takes time. You may need to work with your horse every day for weeks to gain trust. Be patient, always end on a positive note, and the second you make it an issue is the second that your previous work is undone. Be consistent and persistent and in no time you can shower your horse with a soapy sponge followed by a rinsing sponge.
Some horses do just fine with a wet washcloth or small towel to wash the face.
First, a few safety guidelines. I have seen horses flip out and then flip over or slip and fall during a face washing with a hose. This is complicated by crossties. Broken legs and necks can result. Use another person to hold your horse instead of cross ties if you are unsure about how your horse will react to water near his face. If he reacts terribly, remove the offending piece of equipment and do something else that isn't so scary before you adjust your approach and try again. I would suggest starting with a wet washrag, then a wet sponge, and maybe months after that think about using a hose (at a trickle...)
I never assume any horse will be fine with a spray of water to the face. Most want to avoid water in their ears and eyes and probably nose, too! I play things safe. I usually just start with a really damp rag and use that to wash the face, not even bothering with the hose. Elbow grease goes a long way here. For a soaping, I use a very diluted mix of mild shampoo and a sponge to apply, and another clean sponge in a bucket of rinse water to wash the shampoo away. I love the big sea sponges. They soak a bunch of water, so rinsing is quick and easy.
Sponges come in all sizes and shapes - the big ones are great for a rinse.
It's also perfectly acceptable if your horse never gets squirted in the face. Ever. If you live and die by the sponge, so be it.
I will never suggest that you get into a forceful fight with your horse about this. If he raises his head, you will probably never be able to pull it down. The more you fight, the more you reinforce this pulling away and make the washrack a bad experience. Move on and come back later. Reward for any little bit you can. Throw a party. Praise praise praise.
Over time, these positive experiences will make bath time easy for both of you. And I guarantee that you will still be just as wet as your horse.