Just the idea of euthanizing a betta makes me shudder.
But I understand that it can be necessary.
I had to 'euthanize' one of my mice once. It was the runt of the litter, and although it survived until it was almost an adult, it would periodically slip into these bouts of sickness. He'd lose fur, stop eating, lose his balance, and get these odd rings on his tail. After a day or two he'd come out of it, but the last time he got sick he was in a rut for over a week. He got worse than ever before, and his buddies even started to nibble on his tail and ears. One morning I looked at him and realized, "He's not going to make it
." This odd strength came over me, and I knew I had to make sure that his suffering was ended as soon as possible. I couldn't bear the idea of cutting his head off or drowning him, so I screwed my eyes up and quickly broke his neck. It's not a very fond memory, but I am certain that I did the right thing. He was starving and being eaten alive by his siblings. Not much of an existence.
EDIT: A necessary read for anyone who thinks they need to euthanize their fish. Even if you don't need to, it's good stuff to know: http://nippyfish.net/2007/02/12/euthanizing-aquarium-fish/