What might be the reasons for a 6 year old kid to get fear for climbing up or down certain staircases in public places? The kid can climb stairs at home and other frequently used stairs, but gets hesitant either to climb up or down on seeing some wide staircases. Any possible recommendation and solution to overcome this issue.
In my experience, kids tend to lack the ability to judge risks effectively. They don't have the experience in judging risks, and their brain isn't fully developed. In many kids this leads to "risk-taking" behavior, where they do things that are too risky.
But some kids go the other way - they are too cautious. You can help that by finding out why they're scared, and then giving them some experience with the thing they're scared of. Ask why they're scared! Maybe it's the lack of rail on one side (if it's wide staircases). Maybe they saw someone fall. Maybe the stairs are very crowded. Find out why, then help them find strategies to overcome that fear - and go up and down the staircase several times to practice.
My son's reading scores dropped in grade school, and it was due to his eyes not tracking together all of the time. For my son it effected his reading the most, but his eyes were out of sync off and on all day and he just worked around it, he never said anything to me.
You might consider getting your son's eyes checked. If his eyes aren't always tracking together it can effect his depth perception. On a familiare staircase he can use many visual clues to overcome any oddity in depth perception, but on a new staircase, especially a wide one where he can't see both sides, it would be harder for him to understand what he's seeing.
Waiting for his eyes to coordinate and then processing what he sees could both cause your son to hesitate when he comes to a new set of stairs.