As always, if you do not tell us where you live, it is hard to give specif banks - as they might not be in your country / state / continent.
If your boyfriend tries to manage some of his previous debt, it might make it easier to open an account - and if you already have an account,y uo could try opening a joint account in your name at the bank you are already with - and then perhaps a few months down the line (after running the account properly), see what happens if he tries to get one in his own name only - or you ask to come off the joint account.
Providing you make it clear that you do NOT want a cheuqe book, debit card or credit card - just a pying in book for the time being, sometimes banks will consider bad debters.
Alas if they have GOOD reason to think they will lose money, they are not going to be bursting a gut to open an account.
To go back to the first point, trying to sort out some of his bad credit would be a plan - if not a pleasant one.