As mentioned in the question, we know that Muggle money (pounds, and presumably euros, although the latter is not stated) can be traded at Gringotts. Hermione's parents do so in the second book:
“But you’re Muggles!” said Mr. Weasley delightedly. “We must have a
drink! What’s that you’ve got there? Oh, you’re changing Muggle money.
Molly, look!” He pointed excitedly at the ten-pound notes in Mr.
Granger’s hand.
—Harry Potter and the Chamber of Secrets
The goblins of Gringotts apparently put Muggle currency that they receive "back into circulation," presumably in the ordinary Muggle economy:
Those goblins are sneaky people. They manage to put the Muggle money
back into circulation. They are like “fences” -- British slang, do you
understand it?
As such, we can make a very good guess as to how Muggle-born witches and wizards get the funds needed to open a Gringotts account: they simply convert Muggle money.
We have no information, to my knowledge, on what checks Muggle-born students (or any other wizards) need to go through to open an account at Gringotts. However, I doubt there would be any.
Their names are down on the list at Hogwarts, so their chances of pretending to be anyone else for the purposes of fraud are low. Of course, one could use Polyjuice Potion to try to open an account in someone else's name, but the odds of a student not even in Hogwarts doing so are extremely low.
As for how Gringotts keeps track of Muggle-borns after they open an account, it seems as if the Gringotts bankers recognize the key to a person's vault:
"Morning," said Hagrid to a free goblin. "We've come ter take some
money outta Mr. Harry Potter's safe."
"You have his key, sir?"
"Got it here somewhere," said Hagrid, and he started emptying his
pockets onto the counter, scattering a handful of moldy dog biscuits
over the goblin's book of numbers. The goblin wrinkled his nose. Harry
watched the goblin on their right weighing a pile of rubies as big as
glowing coals.
"Got it," said Hagrid at last, holding up a tiny golden key.
The goblin looked at it closely.
"That seems to be in order."
—Harry Potter and the Sorcerer's Stone
At a guess, then, the first time a Muggle-born comes to Gringotts they will deposit Muggle money into their account, and will be given the key to a vault. Presumably thereafter the goblins of Gringotts recognize them by the key (or indeed by their face—if they can recognize a key, perhaps memorizing the faces of the few thousand witches and wizards in Britain is not so hard).
If someone wanted a loan or some other financial instrument where trust was important, Gringotts might want proof that they could pay it back, which might necessitate some sort of magical credit check. However, we don't even know whether Gringotts gives loans. Muggle-borns opening an account almost certainly want to do nothing more than deposit and withdraw money, at least initially, since (a) they are children and thus likely do not have large expenses and (b) Hogwarts tuition and supplies are covered:
"That is easily remedied," said Dumbledore, drawing a leather money-
pouch from his pocket. "There is a fund at Hogwarts for those who
require assistance to buy books and robes. You might have to buy some
of your spellbooks and so on secondhand, but —"
—Harry Potter and the Half-Blood Prince
By the time Hogwarts students would be likely to want loans, they presumably have a fairly established identity within wizarding society.
So Gringotts doesn't need to check their ID, doesn't need to know about (and probably doesn't care about) their Muggle credit history, and likely simply does not care about their background in the Muggle community. As long as they can deposit Muggle money (which can then presumably be exchange for food, raw materials, etc) they have the capital needed to open an account, and the Gringotts employees have ways of keeping track of them thereafter.
Not that most people would dare try to defraud Gringotts anyway:
"Like I said, yeh'd be mad ter try an' rob it," said Hagrid.
—Harry Potter and the Sorcerer's Stone
