I'm asking for a friend, a wizard who until recently worked at the Apothecary in Diagon Alley, but is really interested in banking, like his Muggle cousin who made millions of dollars working for a bank. Anyway, he's trying to stay local in DiagonAlley, and he wants to know if they employ non-goblins before he tries for an interview. Thanks!
Yes. The most famous example of a non-Goblin employee at Gringotts is Bill Weasley.
It's referred to several times, but I think the first reference I can find is in Chamber of Secrets chapter 4:
Harry had never met either of them, but knew that Charlie was in Romania, studying dragons, and Bill in Egypt, working for the wizards’ bank, Gringotts.
And fret not if curse-breaking isn't your friend's forte, for they also have desk jobs, which Mr. Weasley transferred to when moving back to Britain during the second struggle against Voldemort.
From Order of the Phoenix, also chapter 4:
‘Is Bill here?’ he asked. ‘I thought he was working in Egypt?’
‘He applied for a desk job so he could come home and work for the Order,’ said Fred. ‘He says he misses the tombs, but,’ he smirked, ‘there are compensations.’
I consider the dragon a non-goblin employee of Gringotts. While it might not be a very well treated employee, I've seen some companies treat their employees worse.
Gringotts offers several jobs opportunities to non-Goblins, but they may not help your friend so much.
They say there’s dragons guardin’ the high security vaults
(Harry Potter and the Philosopher's Stone, Chapter 5)
GRIPE WITH GRINGOTT’S BANK
Am I alone in wondering why Gringott’s Bank has recently (...) I was taken aback to find one of these creatures standing guard outside my own vault on my last visit. (...) As I failed to answer every one of the sphinx’s riddles, I was unable to get past it to my gold.
-- Mallory Twiddle
(Daily Prophet 1, "Gripe with Gringott's Bank")
Are you seeking a challenging career involving travel, adventure and substantial, danger-related treasure bonuses? Then consider a position with Gringotts Wizarding Bank, who are currently recruiting Curse-Breakers for thrilling opportunities abroad…
(Harry Potter and the Order of the Phoenix, Chapter 29)
“Is Bill here?” he asked. “I thought he was working in Egypt?” “He applied for a desk job so he could come home and work for the Order,” said Fred.
(Harry Potter and the Order of the Phoenix, Chapter 12)
Needed by Gringotts Bank. No qualifications necessary. Fireproof robes and balaclava provided. Apply in writing giving names of next of kin. Salary: 7 Galleons per week.
(Daily Prophet 2, "Classified: Jobs")
HOW FAR WILL FUDGE GO TO GAIN GRINGOTTS?
Cornelius Fudge, the Minister for Magic, denied that he had any plans to take over the running of the Wizarding Bank, Gringotts, when he was elected Minister for Magic five years ago. Fudge has always insisted that he wants nothing more than to ‘co-operate peacefully’ with the guardians of our gold.
BUT DOES HE?
Sources close to the Minister have recently disclosed that Fudge’s dearest ambition is to seize control of the goblin gold supplies and that he will not hesitate to use force if need be.
“It wouldn’t be the first time, either,” said a Ministry insider. “Cornelius ‘Goblin-Crusher’ Fudge, that’s what his friends call him. If you could hear him when he thinks no one’s listening, oh, he’s always talking about the goblins he’s had done in; he’s had them drowned, he’s had them dropped off buildings, he’s had them poisoned, he’s had them cooked in pies…”
(Harry Potter and the Order of the Phoenix, Chapter 10)
Of course not all of these jobs would be the best for your friend. He may not be a Dragon, Troll, or Sphinx. He may not be cut out to be working abroad curse-breaking. Your friend may dislike boring desk jobs, or to afraid to feed dragons. In truth Gringotts saves their best jobs for fellow Goblins. However this could change. You may want to show your friend this flyer.