** Spoilers ahead for the book Harry Potter and the Deathly Hallows **
Ariana Dumbledore was attacked when she was six years old, which left her severely traumatized and unable to control her magic. Her family decided to hide her condition. The father, Percival died in Azkaban without a word of explanation why he suddenly thought it'd be a good idea to assault three muggles, never telling anyone that it was revenge for those people hurting his daughter.
Adriana's mother, Kendra moved the family to the village Godric's Hollow, where Ariana was locked up in the house, not allowed to be seen by anyone, except the family. At age fourteen, Ariana accidentally killed Kendra with one of her bursts of uncontrolled magic.
But even after the accident her brothers, Albus and Albeforth continued with a course that proved ineffective instead of seeking professional medical treatment.
As for why, this is Albeforth's explanation:
"... because if the Ministry had known what Ariana had become, she’d have been locked up in St. Mungo’s for good. They’d have seen her as a serious threat to the International Statute of Secrecy, unbalanced like she was, with magic exploding out of her at moments when she couldn’t keep it in any longer. “We had to keep her safe and quiet. We moved house, put it about she was ill, and my mother looked after her, and tried to keep her calm and happy. “~ Harry Potter and the Deatly Hallows - The Missing Mirror
Why the Dumbledores thought there was need for such drastic measures and sacrifices? It isn't as if St. Mungo resembled the infamous muggle mental hospitals, shuch as the London Bedlam. Exactly the opposite, from what we know from Harry's visit to the place, the patients are taken care by competent staff. They found quickly enough an antidote to Nagini's venom to treat Arthur Weasley, Lockhart seems content enough and his healer is a very nice lady, nothing like nurse Ratched:
A motherly looking Healer wearing a tinsel wreath in her hair came bustling up the corridor, smiling warmly at Harry and the others. “Oh Gilderoy, you’ve got visitors! How lovely, and on Christmas Day too! Do you know, he never gets visitors, poor lamb, and I can’t think why, he’s such a sweetie, aren’t you?” ~ Harry Potter and the Order of the Phoenix -Christmas on the Closed Ward
We also know that there are no obstacles visiting the patients: Alice and Frank Longbottom are regularly visited by their son, Neville and by Augusta Longbottom, and the healer thinks it's a pity no one visits Lockhart who is located in the closed ward:
“Listen to him,” said the Healer, taking Lockhart’s arm and beaming fondly at him as though he were a precocious two-year-old. “He was rather well known a few years ago; we very much hope that this liking for giving autographs is a sign that his memory might be coming back a little bit. Will you step this way? He’s in a closed ward, you know, he must have slipped out while I was bringing in the Christmas presents, the door’s usually kept locked . . . not that he’s dangerous! But,” she lowered her voice to a whisper, “bit of a danger to himself, bless him. . . . Doesn’t know who he is, you see, wanders off and can’t remember how to get back. . . . It is nice of you to have come to see him —” ~ Harry Potter and the Order of the Phoenix -Christmas on the Closed Ward
So the patients are treated competently and nicely by the staff, not isolated completely. Even if there wasn't any hope for recovery there are many advantages of being in a hospital with a qualified medical personnel, people who could whip up a calming draught or research other potions to treat her. They probably also have medical records of similar cases - it's very unlikely that Ariana was the first kid showing magical abilities who survived an attack by muggles.
Albeforth claimed he was the one who had the most calming influence on Ariana, but by his own words, he was the one who (with the best intentions) started a shouting match in a presence of a traumatized and unstable patient, one that escalated into a fight which triggered an episode. As a result Ariana was killed by accident, so I am not so sure he qualifies either as the most competent person to look after her.
So why was the family so opposed to treatment in St. Mungo and kept an unstable patient who was danger to self and others at home? The deaths in the family could've been avoided, including that of the patient. Seems very unreasonable to me...