In the UK it seems to be the exact opposite; they don't particularly care to have you as a customer unless you can meet some threshold of worthiness. Anyone know why there's such a difference?
I think there are several things going on there. Sign of the times could be one, like what DrSuperL99 said. Branch staff aren't always as knowledgeable & well trained as they could be. Also what banks advertise (what they say) and what they do are often completely different things - unfortunately. If you review this link (what banks have published about themselves - British Bankers Association):
http://www.moneymadeclear.org.uk/pdfs/bank_accounts.pdfYou can see that for
basic bank accounts, as long as the customer has provided sufficient proof of identity - really the only reason they should be turning you down for a basic account are one of the disqualifying factors listed - usually undischarged bankrupt, or record of fraud, which don't apply to most people.
Therefore if the bank is saying you can't have a basic account, a customer should ask why & be given the reason. Be polite but persistent. (Heck - take that leaflet with you!) If the reason doesn't seem legitimate, then ask someone higher up until you are satisfied (or more likely, until you get the basic bank account). Wave the leaflet around, point out what they have published about their own policies...

The thing is that banks don't make money off a basic bank account customer. Therefore, the bank
assumes you want a higher level account - with overdraft, with cheque writing (which is going away, btw...note cheque writing is considered an extension of a credit facility here), etc - because (bonus for the bank!) they can charge you fees on these (often a monthly maintenance fee at the very least, when you're not even owing them money - rather you are paying them for the privilege of their keeping your money for you). When you go into a bank & say you want an account, they assume this is what they are going to set up for you - credit checks are run, so on & so forth, and this is the hurdle at which new folks often fail - not because you have bad credit, but because you have no credit, here. And then maybe the bank staff doesn't want to spend anymore time on you to get you into the basic account = little to no profit for the bank. (wonders if staff targets play into this somehow? commissions - etc?)
However, if you know the name of the bank's basic account product (Barclays Cash Card, Cooperative Cashminder, Halifax Easycash, Lloyds Cash Account, etc etc etc) - go in & be assertive and make it very clear that is what you want...well the bank has less leeway to get the facts confused....

(although it could still happen in application process matters, proof of identity, etc)