"Be bothered about (something)" is the verb "bother" used in the passive voice. "Bothered about" isn't a common collocation, it would usually be "bothered by." It's much more common to use the active voice and "(something) bothers me or "(something) doesn't bother me." It would sound much more natural to say, "Falling doesn't bother me." You might also hear "bother with (something)" or "bother to (do something)", but they don't mean the same thing as the others. When it comes to collocations, it's always better to follow the examples you hear natives using than to try to come up with new ones on your own.

‘To bother about someone’ is a colloquial phrase meaning ‘worry about someone’. It’s often used in the negative imperative. In my circle it is universally understood and not uncommon. For example: A. I’m worried that being vegetarian you won’t have anything to eat. B. Don’t bother about me. I’m not going to starve! A. I’m sorry that you can’t stay with us, Grandma and Grandpa. Our place is just too small. B. Don’t bother about us! We found a decent hotel in town. We’re just happy to be able to see you after so many years. Used positively, it’s less common. Maybe I’d use it to refute the negative imperative. A: Don’t bother about me! (Don’t go to any trouble) B: I’ll bother about you if I want to!

One is a verb and the other is an adjective. That's all you need to know. I hope these examples help you understand. Just one warning: the word "bothered" can play two roles. It can be the past tense of the verb "to bother" and it can also be the past participle (p.p.) of the same verb. In this discussion "bothered" is being used only in the sense of a past participle acting as an adjective. ---ADJECTIVE "BOTHERED"--- As an adjective, "bothered" can describe any noun but normally that noun is a living thing: "Bothered by his sister, Billy asked her to stop talking". (adjective describes Billy) "Billy is bothered by his sister's talking" (adjective describes Billy) "Bewitched, bothered and bewildered am I" (adjective describes "I". This is a line from a famous song.) "Mary is annoying Billy. You know it from his face. He looks like a bothered boy" (adjective describes "boy") "The English teacher, bothered by the word 'get', is fed up and wants to do something else with his life." (adjective modifies "teacher") "The editor says he is bothered by the word 'impactful'. He says he will scream if he hears it one more time". (adjective modifies "he") ---ACTIVE VERB "BOTHER"--- Your sentence "I don't bother about falling" is poor. It is not even clear to me what it means. Here are some valid uses of "bother" as a verb: "Billy's little sister bothers him, so he moves away from her." "That lady bewitches, bothers, and bewilders me." "The word 'get' bothers every English teacher." "The word 'impactful' bothers editors. Indeed, it can drive them insane."

Being bothered about something is almost always used in a negative sense : ’I’m not bothered about falling' - I’m not worried / concerned about . . 'I can't be bothered to do . .' means I’m not going to do it 'I don't bother about . .' means the same thing. 'I can't be bothered to correct their spelling mistakes.' 'I don't bother (about) correcting their spelling mistakes. 'When referring to people [or possibly animals] : 'Is he bothering you?' means 'disturbing you / annoying you / inconveniencing you / getting in your way’ Tbh I think it's a slightly old-fashioned word, probably not used as much now, but it's a lot more polite, and therefore safer, than some of the more vulgar expressions you may hear.

like "frighten" and "be frightened" action and feeling