Here is my take on giving out banking information. Every time I write a check I am handing out my bank name, routing number and account number. The bank is now responsible for not handing my money over to unauthorized folks, even if they provide my account number.
I add a layer of security. I have a savings account and a checking account at the same bank. The bank is not authorized to move money between accounts to 'cover' demands, though I can do so on line myself. I keep a small balance in the savings account just to keep it open and the 'banking information' I give out is to the savings account. When money comes into the savings account I transfer it promptly to checking, leaving only the small balance to keep it free.
I have had no issues with anyone trying to empty my checking or savings accounts.
I am more concerned about handing out my Social Security number, but without doing so I will not be able to work.