Without knowing where in the world you are it's hard to name any specific institutions but here are some things I've learned over the years:
1. Don't do business banking virtually. By having a relationship with my local bank officer I was allowed to go over my line of credit limit by $5,000 for 14 days on her authority alone. The only reason she let me was our personal relationship. People matter.
2. Ask if you'll automatically be offered credit limit increases based on account activity or if you need to apply each time you want to expand credit. Applying each time will likely create a 'hit' on your credit score. I once had my credit expanded by $20,000 simply because of the account activity.
3. Get a clear understanding of the fees and understand that dealing with you costs the bank money. I'm very suspicious of 'free' accounts of any kind. There is always a cost.
4. Make sure the branch is convenient. I don't like waiting for an hour on the phone to speak to someone in a call center. When there is a royal screw-up in your banking nothing beats marching down and talking to the commercial manager. They'll want you out of their office fast and that means getting a resolution most of the time.
5. If you're going to be doing business internationally, ask about foreign branches or brands. Here in Canada most of the banks own US banks so it can make things easier for business in the US to open a US account in the USA from the comfort of home.

Set up a call if you'd like to talk about your specific needs.

David C Barnett