If you do an image search for "old checks" you'll note that until about the 1960s there was no account number on bank checks, and when they appear they are invariably the magnetic OCR type numbers that are obviously for machine reading.
If you write me a check in 1945, say, and I take it to my(?) bank to deposit/cash it, how does the bank know whose account to debit? All they have to go on is the signature at the bottom, with variable legibility and lots of room for ambiguity (common names, first initials, etc.).
It was really just a matter of scale. Over time, banks have consolidated. Banks today are much larger and operate many different branches. They're often regional or national. In the mid twentieth century, you would be much more likely to use a truly local bank than you are today. Today, many people prefer larger banks because of conveniences like national ATM networks, so there is some benefit to banking with a large institution.
With a smaller bank, you can use security measures that simply don't work on a larger scale. Say you live in a town of 10,000 people. Maybe there are two or three banks in that town. You don't need to order things online, and most of the things you buy will be from small local shops.
The merchants at these stores know these three banks. They will accept checks from these three banks and these three banks alone. They won't accept out of state checks.
Say you want to pay me $20. You have an account with one of these banks, so I am willing to accept your check. I bank at a different one of the three banks. When I deposit your check, the bank takes the check to your bank and orders a withdraw.
How does your bank know that the check is legitimate? Simple, they have a relatively small number of customers. They might only have 1,000-3,000 accounts. If your check has your first, middle, and last name on it, it's highly likely that you're the only person at that bank with your exact name.
How do they make sure the check is valid? Well in their files, they keep a copy of your signature. When a check comes in, they simply compare the signature on file to the signature on the check. If the signatures aren't the same, the check is rejected.
That was really all they needed to do. A lot of this was also helped by interpersonal networks and shop owners and banks knowing all of their customers by name. A shop owner might have a policy of only accepting checks from someone who is a regular customer.
Today, we're used to having much stricter security for banking, but this is only really necessary because of how national and global commerce is now. Ordering products from a continent away with an account from a national bank simply requires more complicated security techniques than local banking and local commerce.
In addition to the answers above you may find some information in the previously asked question, http://www.reddit.com/r/AskHistorians/comments/1u0ggy/how_did_cashless_point_of_sale_transactions_work/