Most of the time we fry brain and eat it. But due to its semi-solid state, it doesn't seem good.
Also Wikipedia says:
Consuming the brain and other nerve tissue of animals may be hazardous to health.
How can I cook brain and avoid both the semi-solid texture and the potential health hazards?
As far as I know, neither one is possible.
The brain is supposed to be very soft and creamy when cooked. It is prized for this texture, there are languages where the idiom "it is soft as brain" takes the place of the English "it melts on the tongue". All recipes for brain are written to make it softer, not firmer.
Physiologically, brain cells are very different from muscle cells. Most of the cells in the brain have the purpose of electric insulation, and are practically sheets of myelin. Myelin is mostly fat and water, with a little protein thrown in. Trying to make it firm is akin to trying to make butter firm. It will always stay creamy, even if you cool it down to a point with little give/smearability.
As for safety, the brain has roughly the same dangers as eating other types of meat, plus the danger of prion caused disease. The danger it has in common with other meats are averted by following standard cooking practices, as with all other meats of the same animal. I doubt that any source warning you against eating brains will have these dangers in mind.
Prion caused diseases are a different matter. They are very rare, but very dangerous (CJD, commonly called "mad cow" in the media, is a prion disease). And no matter what you do to the brain, you can't destroy the prions. They survive at temperatures at which the tissue itself is incinerated to unrecognizable ash. You don't have a chance of destroying them by chemical means either, if you want to keep the brain meat edible. If you happen to eat a contaminated brain, you will be exposed to active, pathogenic prions, no matter how you cooked it. As far as I know, you can't avoid getting the disease after prion exposure (although I'm not entirely sure about that part). There is no cure for it.
Despite the popular name, the prions causing CJD can be found in the brains of animals other than cattle. This is why the Wikipedia article on monkey brains contains the sentence you cited, the prions can be found in monkey brains too.
