My 7 year old seems to be afraid of a lot of playground equipment. It's not the height so much as I believe that he's afraid of falling and getting hurt. (He'll climb up tall stairs and go down tall slides, run across high bridges...) We were at a playground recently where they had a wall climb, monkey bars, a mini zipline, hand over hand bars (I never learned what those were called, a twisty metal pole to climb and one of those rolling logs with the bar to hand on to. He wouldn't do ANY of it. He'd go to each thing like he was going to start, then he'd say that he was afraid and what if he fell?? I tried to reassure him that I was right there and would catch him and help him, but he was still really freaked out. How do I help him acclimate and learn to overcome his fear? He WANTS to, and I can see that he's very frustrated by his own feelings.
Acknowledge his fear. "Wow, yes, it's huge isn't it? And it's pretty scary."
Find out what result he is scared of - you've said it's being hurt if he falls, so check it is actually that.
Ask him what he could do to avoid that happening. Then ask him if he can think of examples of risky things that he does safely or other things he does to keep himself safe (wearing a helmet when he rides a bike?) Remind him that there are things that are still a bit too dangerous for him - boiling a kettle unattended, or going high up a climbing wall without a rope - and then ask him if he thinks you'd allow him to do those risky things.
Start small. Find a smaller bit of play equipment that he's comfortable with. Reward good behaviour. Model good behaviour too - try the equipment yourself.
After he's tried it ask how he feels it went; if he felt safe; what he'd change; then finish with "Was it fun?"
This answer very loosely uses concepts from "cognitive behaviour therapy". There are emotions (fear) that are driven by hot thoughts ("I'm going to fall off and hurt myself") supported by flawed evidence (you need to find out what the evidence is). Your aim is to stop the hot thought happening, and to allow him to correct the flawed evidence with something more realistic, and to reduce the fear.
He should be doing most of the work - this process is not you telling him things but you helping him to discover these things himself.
Here's some information from the English NHS about anxiety, including anxiety disorders: http://www.nhs.uk/Conditions/anxiety-children/Pages/Introduction.aspx
I would take a step back from the question and ask two (2) things:
Why does the kid need to play on the equipment? Do you want him to for social reasons (so he's not left behind, etc), physical reasons (this will teach him dexterity that will allow him to do x, y, z activity you have planned for him), or a psychological reason (physical confidence will make him confident in other areas in life)? This is a worthwhile distinction because the issue is (probably) not the playground per se, but some more profound value of yours that the playground stands for.
This is mentioned in other answers, but what's he really scared of and what might have caused him to be scared of it? Keep that in mind and take baby steps. Again, this is explained in other answers.
IMO, I say ditch the playground and go do something else. No need reinforcing inadequacy, as he probably feels like a bozo watching all the other kids jump like trapeze artists. Go do something that makes him feel strong, something that he does fearlessly, something that excites him. And let him do it. Ask him to show you, ask him how he does it and why. See where I'm going with this?
ALSO, this is totally projecting, but I would say the best thing you can do for the kid is get out of his way. Don't follow him around the playground. He probably doesn't need a friend; he needs a mentor to challenge him and support his growth. If he asks you "What if I fall?", laugh and smile and say "I don't know, buddy." And if he says "I'm scared," tell him the truth: "Dude, I'm scared sometimes, too. But that's never a reason to not do something". I'm not saying ANY of this is easy, nor possibly to do perfectly. But it's 100% true in my experience that the source of most kid's neuroses and issues are their bumbling parents trying to protect them from the world.
Good luck and keep being an awesome parent! Your little dude is lucky to have you.
In addition to the other good answers here I would counsel patience. Let him go up to the equipment and then back out. Let him do it lots of times. Eventually he will overcome his fear. But this only works if you let him do it in his own time. If you get impatient and push him into doing something he isn't ready to do then you will just reinforce his fear.
Also, you might try backing off the "I'll catch you". Instead sit back away from the equipment and let him run around and explore on his own. Having you hovering there ups the pressure.
A couple of anecdotes; I went through a phase of climbing up a slide, deciding it was "too way down", and going back down. My son did something similar with a spiral slide in our local playground; he spent months going to the top, sitting there, then coming back down.
Another thing my son did was to decide that only older children could do things. So on his 5th birthday (if I recall correctly) he announced that he was now old enough to ride his bike without stabilizers. And then did exactly that. Maybe you could plant a similar idea.
I would suggest you do some research on risk taking in children. Especially in recent decades, parents often unintentionally stoke this sort of fear by doing things like being "right there [to] catch him and help him." If you let them occasionally fall from a short height when they are three years old, they will be a lot more confident on the higher climbing walls when they are seven.
If you're always there to catch him, that shows him you don't trust his climbing ability. If you don't trust his climbing ability, why should he? Instead show him an example of fearlessness by either by going first and doing it yourself and expecting him to follow, or by removing yourself beyond arm's reach and looking unconcerned.