Who gets Muggle-born students onto Platform 93/4? Their parents can't pass through the magical wall (I suppose).
Harry only gets onto the platform by good fortune in his first year after bumping into the Weasleys. How do the rest of the Muggle-born students access the platform?
Well, I do remember that in the Chamber of Secrets, Hermione's parents were in Diagon Alley with her. That clears the air around muggle-born students' parents being able to be in the magical world.
And remember how in the same book,
Dobby sealed the opening!
Maybe it has a special opening for muggle parents?
There's probably some kind of way that muggle parents are granted a special permission so that they are able to go with their children.
EDIT:
Also, the seventh book talks about
Lily and Petunia having an argument on Platform 9 3/4 (in Snape's memories).
Petunia is a Muggle, yet she was on the Platform! Lily and Petunia's parents are also mentioned later.
So yes, it is possible, but I think you may have to go through with a witch or wizard if you are a Muggle. There must be some restrictions, otherwise muggles would have discovered the Platform by now!
Well, it seems that a teacher would be sent to explain the nittygritties and to deliver the "Congratulations! You can do magic!" letter to Muggle-borns.
Note how Dumbledore went to invite Tom Riddle (at that time Tom was just another Muggle born) and Hagrid (not strictly a teacher at this point) invited Harry (not strictly a Muggle-born, but he lived with Muggles).
I don't think the barrier keeps Muggles out at all.
Given that the students are told to run at the barrier, I think it's probably enchanted so that if you touch it slowly or gently, it feels solid. It only yields if you move through it quickly. So the Muggle parents are simply given the same instructions as all the other parents, and perhaps are helped by wizarding families.
Picture this: You're an average Muggle bystander at the station. You happen to see someone run toward a barrier, and go right through it and vanish. Are you likely to run at it yourself to see what happens?
No, first you'll walk over and put your hand on it...and it'll feel perfectly solid. You'll walk away and make an appointment to get your vision checked.
If you did happen to run at it, you'd get through. But an ordinary person wouldn't do that.
First year muggle-born students always seem to have their supplies, owls, wands, et cetera on the first day of school before boarding the train which means they had to be in Diagon Alley first.
As first years are usually 11 years old this means the parents/guardians must be able to go into the magic world with said child to get to Gringotts and exchange money to pay for supplies and later say goodbye and make sure they board the train with everything they need.
This means that somewhere along the way, either through instructions in the acceptance letter or through a Hogwarts staff member's visit they were given some type of advice/information on how to get into the wizarding world through the Leaky Cauldron and then onto platform 9 3/4.
Since the muggle born kids do not yet have wands, it only makes sense that some kind of wizarding adult is there to let them and their parents into Diagon Alley through the brick wall at the Leaky Cauldron and to tell new muggle born students how to get onto the platform once the day arrives. Since they now have their wands in their possession the barrier opens for them and the people with them.
I seem to remember parents staying very close to their children as they ran through the barrier. The possession of a magical wand also explains why normal non-magic folk cannot get through the barrier.
Firstly, I am not convinced that muggle parents of invited wizards could not get through. As I understand it, anyone could get through when the gateway is open, but most people don't try running at walls.
However, the parents don't have to get through - only the children have to. Now how they know to run at a certain wall we are not told. It may be that more attentive parents than Harry's will ask these questions before their precious child has to travel to school.
This is utter guess work but I think Muggle-born's parents have a magical permission of sorts. In other words, if your kids or siblings are going to Hogwarts you can pass the magical barriers in Platform 9 3/4 and Diagon Alley by default.
Harry does not get in by accident. When the Hogwarts Express starts, there are bound to be lots of wizard parents or older students in the railway station. Surely many of them will keep an eye on any first-years arriving.
Muggle-borns get to the platform as there must be a separate kind of letter of acceptance which tells them how to get through Platform 3/4 but the most likely thing that must happen is that as Snape suggested in book 7 that someone from the school comes to explain the muggle borns about the magical world and how to get through to it.
I would presume that since the Muggle born student obviously has enough magic to attend Hogwarts and to get through the barrier, his/her parents and siblings would have just enough magic in them from their ancestors to get through the barrier themselves.
As JK Rowling stated in a web chat on The Leaky Cauldron:
Muggle-borns will have a witch or wizard somewhere on their family tree, in some cases many, many generations back. The gene re-surfaces in some unexpected places. (Leaky Cauldron web chat)
My guess that the magic put on the barrier allows someone with just a little bit of magic in their blood to pass through. Unlike other Muggle prevention spells (Quidditch World Cup, Hogwarts, etc.) which probably have strong Muggle barriers.
The easiest answer in a world of magic would be to have a normal seeming door enchanted to only pass people with a Hogwart's invitation or a special pass that is included with the Hogwart's invitation. Without that item, the door remains "stuck" and will not open. Furthermore, marking the door with signs of "This door is blocked." and "Not an exit" would have all Muggles understand that the door does not work.
So the Muggle parents would have directions to the door and instructed to bring along the special pass. They'd find the door works for them (probably loud, a bit hard to open, and creaky), and that would give access to the Platform 9 3/4s (and any other magical platforms the wizard world has there), and that would seem reasonable to them. They'd only find that the door is stuck and won't open if they return without the parental pass.
Of course, with magic, it doesn't need to be reasonable, but reasonable helps the Muggles from freaking out, and considering all the Muggle born and Muggle raised wizards there should be a few that have worked out a way for purely Muggle families to get to the platform.
I don't remember where I read it but apparently a representative from the wizarding world comes to inform the muggle born when it is time to go to hogwarts if they come they probably stay and guide them and the family
