For a BACS/Bank Transfer you do not need to give anyone any information that isn't already provided, for example, account number, sort code, name and address <as this is required to dispatch an order anyway, but for a BACS/Bank Transfer, as a seller, you do not need to know anything else like mothers maiden name and all that security stuff that the bank asks its customer. You just need account number and sort code, valid from date in some cases or just expiry date. Me as a seller, I can't do anything with that information except bill the person's bank. Ok, the risk of fraud is still there, like with anything, as I could bill for more, which I wouldn't do anyway, but as the buyer, you'd be checking your bank statement anyway, or should be out of common sense, to check that I have billed for the correct amount!
Paying by credit/debit card is the same. You have to provide the long number, name on card, expriy date and/or valid from date. That information is needed to bill the card, and to make sure it's being billed to the right person etc.
Identity theft is not anything new. It's been around forever. Well, ever since the introduction of paper bank statements that people throw out in the trash and people leaving certain details open to attack anyway because they're careless. It's just that today, there is more coverage and awareness of it because of fraudsters making up bogus emails and links to bogus websites to get your login information to act as you, pay for stuff in your name or clear out your account.
So a BACS/Bank Transfer is safe and secure as using any other method of payment, when even then, all those other payment methods are still vulnerable to threats too.
And as an experienced and trained geologist who has specialised in planet mechanics, volcanoes, earthquakes, tsunami, ancient weather, mega storms, ice ages, sea level rise and drops over thousands of years, solar system, planets, asteroid and commets etc, worked on proving certain biblical stories to be based in fact due to geological evidence of them, like Noah's flood occured at the end of the last ice age during a global warming event which saw oceans around the world rise by a few hundred feet which has now been proven but whether the story of Noah himself is ture is a different matter altogether, ten plagues of Egypt caused by a major volcanic eurption at the island of Santorini which Moses was nowhere near the red sea but was in fact at the sea of reeds which was more marsh land, could be got across but because of the Santorini island collapsing into the sea when the volcano erupted, several tsunami waves hit the coast of Egypt where moses "parted" the sea of reeds (misinterpretation of a word for red/reed, led to the wrong place which is why no one could find any archaeological or geological evidence of the parting of the seas and Moses's people because when you part a sea it leaves a geological fingerprint and when you are moving hundreds of people across a land, there has to be evidence of it, burning from fires, animals bones etc from food eaten, and the red sea is clean of such things but the coast of Egypt has traces of tsunami events and ancient land reclaimed by the sea), but I can safely say there is no chance of the Earth crashing into the sun, however, when the sun enters the later stage of its life in like 5/6 billion years time, humans most likely won't be around then anway, the earth will turn into a red giant, and will expand in size, swallowing mecury, venus, earth and mars, before going supernova, destroying our solar system for which gasses and debris will be all that's left and out of that will form another solar system and the circle of life/evolution will continue. So we're quite safe for the time being..