yes your details will be ok on a site as it is only an account number and sort code. if you are really that worried you could yuse
www.paypal.com they can take the payments for you, most people that have a computer will have an account or will know about paypal paypal is a protected payment gateway
you could always get your customers to pay via standing order this is what we do, along with paypal and direct into bank account via post office or banks