A thief cannot gain access to your account just on the basis of having your routing number because this number identifies the bank as a whole rather than your account. However, a savvy thief could try to access your account by using your routing number in conjunction with other pieces of information.
Routing numbers were first introduced by the American Bankers Association in 1910. Banks use routing numbers to identify particular banks in the same way that a bank uses account numbers to identify its own customers. When banks receive check deposits, bank employees use the check routing number to identify the bank that the check was drawn from.
Banks also use routing numbers to send wire transfers. When you send money electronically, your bank uses the routing number to send your money to another bank.
Routing numbers are not private and you can find out any bank's routing number by checking one of the many online directories or by calling a bank directly. But having a routing number in isolation means nothing to the thief. However, if the thief knows that you have an account at a particular bank, then the thief can begin figuring out ways to infiltrate your account.
Legally, bank employees are not allowed to share account information unless they can verify the identity of the person. Identifying someone over the phone can prove difficult and while some banks issue you a password, other banks verify your identity by asking personal questions. A thief could easily search public records to find out your date of birth and address, and then call the bank pretending to be you. The thief could then say, "I am setting up my payroll direct deposit and have my routing number but forgot my actual account number." Providing snippets of personal information may dupe some bank employees into revealing more information to the thief. Some thieves call several different bankers and piece together small bits of information from each one.
While a thief could use your routing number as the starting point in an attempt to gather sufficient information to steal from your account, there are more direct ways for tech savvy thieves to steal your money. Someone with the necessary skills could hack into a retail store's computer and steal your debit card information or steal your mail to find out your bank account number. However, you are typically covered against losses that stem from fraud.