I'd treat a new acrophobic hiker the same way I would any other new hiker
I was out with.
I start out by asking them what they are comfortable with before we even start, as well as what their fitness level is.
For instance, I have a friend in poor shape with terrible balance who occasionally comes with me hiking.
I choose routes with her that are much easier than I would do alone, and that mostly don't involve rock scrambling or unbridged stream crossings.
But we've done a little scrambling anyway sometimes, and she's always been happy to learn she can.
Along the way, I check in with a new hiker, especially when we get to a situation that's likely to be a bit tricky.
In this case, it might mean a rocky climb or a trail with a drop-off.
With a hiker who has no specific problems, it might mean asking how they are doing on a tough, steep section, or just checking on their energy and hydration levels after an hour and taking more frequent breaks than I usually would.
With an acrophobic friend, I'd ask them about what sort of situation makes them most uncomfortable, then mostly avoid it.
This would probably include scrambling, steep switchbacks, and cliffside trails, but it's their call.
No need to avoid all heights, since they already know there will be some.
The point is to give them an enjoyable experience so that they want to come back, and maybe push their limits further next time.
outdoors.stackexchange.com/users/3261/Karen
I have never liked heights, but I have still done a lot of things because I wanted to get to the top more than I was afraid of heights.
I think that the best thing to do is to let this friend decide on where he would like to go, and let his motivation decide how much risk (real or perceived)
you take.
It also helps with the fear of heights to get used to it, for me that has always been hanging belays where you hang around at the top of the cliff for a while and eventually you get used to it.
outdoors.stackexchange.com/users/8794/Charlie Brumbaugh
The other answers have covered the important ground, and are all things that have helped me, so I'll just add a few things from my own experience.
I have acrophobia, a balance disorder, and a number of physical injuries and illnesses that make it hard for me to go places and see things.
My husband, and the only other friend I trust, have found ways to help me do things I only dreamed of enjoying, so I'll suggest a few. Hold on to your friend when they get scared.
That may mean taking their hand, hooking arms, or wrapping your hand around their waist.
Sometimes my husband just holds my coat or sleeve, and I feel better.
Also, depending on the terrain, stand behind them so they know they can step or tip backwards and not fall over.
A hand on my back really increases my confidence.
Some people with phobias just need to know they're not stuck, so remind your friend frequently that you'd be happy to take them back at any time.
That can mean returning to a place they were comfortable and resting a bit, as others have said, or even going all the way back to the car or place you began.
For me, that reassurance has often given me the strength I needed to keep going.
There have been times when I've had to force people to cut the journey short, and been afraid I disappointed them.
I know you don't want your friends to feel that way, but just in case, let them know you've had a great time being together and watching them enjoy themselves, and that you'd love to take them again.
Recently, I went to Schoodic Point , which is a rocky section of the Acadia National Park in Maine.
We stayed where it was fairly flat, without much climbing and no scrambling, but the terrain was rough and scary.
It was something I thought I could never do, and planned to stay in the car.
With my husband and friend using these methods, and what's in all the other answers here, I went much farther than I thought I would, loved almost every minute, and am looking forward to going back!
I look at the pictures from that vacation and still can't believe that was me up there!
Caring for your friend in this way can make this kind of huge impact on their life!
outdoors.stackexchange.com/users/5547/Sue Saddest Farewell TGO GL
As an acrophobic myself, you need to be very calm, helpful and be prepared to turn back and/or abort the hiking.
Hiking could be very helpful to help your friend deal with his fears.
A couple of years ago I went to Itatiaia National Park
and it was very hard to deal with the height.
My friends where more experienced and helped me a lot with the hardest parts.
Before the finish, I decided that was enough for me, and I waited for the group to finish and come back.
I write about this experience here (many photos), but it's in portuguese.
outdoors.stackexchange.com/users/8948/Rodrigo Menezes
Over here we refer to height as "exposure" and we encounter a lot of it in the form of cliff traverses.
In order to take care of a person with some fear, one important thing is to recognise that different people have different limits and comfort zones.
For one person walking on an edge with a 2 meter drop is too much, for another 20 meter is the limit, for another walking on a 200 meter drop is not a problem.
So take it slow and be aware of the other person and when they seem to become uncomfortable.
Sometimes they may want to push their own comfort zone boundaries and "try again" after a minute, sometimes you may choose to find a way around the obstacle.
Ask them what they want to do!
outdoors.stackexchange.com/users/8248/Johan
There is some trouble. One of my close friends has acrophobia (fear of heights). But he wants to go for a trek for what it offers than just the mesmerizing view from the top. How can I take care of one such person?  Obvious assumptions: Hike doesn't include any climbing activity, not even scrambling.
How to take care of an acrophobic person while hiking?
outdoors.stackexchange.com
<hiking><safety>
outdoors.stackexchange.com/questions/14254
outdoors.stackexchange.com/users/2303/WedaPashi
Stay very calm, helpful and be prepared to turn back and/or abort the hiking.  Ask them about what sort of situation makes them most uncomfortable, then mostly avoid it. Depending on the terrain, stand behind them so they know they can step or tip backwards and not fall over. An important thing is to recognise that different people have different limits and comfort zones. Some people with phobias just need to know they're not stuck, so remind your friend frequently that you'd be happy to take them back at any time. Allow them to decide what they want to do, give them some physical contact for reassurance. You can stand behind them and stay calm.
Being calm, standing behind an acrophobic and reminding them they’re not stuck are good points. Allow them to decide what they want to do, give them some physical contact for reassurance.