Other people might argue about the necessity of having a physical bank to have access to, but I literally have used a physical banking location for nothing in about 10 years now. I recommend Ally Banking as an excellent online only option, you can set up a savings and checking account through them and their interest rates blow the vast majority of other banks out of the water. They also have home and auto loans if you need to apply for those although I honestly can't vouch for that as I've never applied for an auto loan through them.
Ally is great. I personally use Simple because it's slightly greater than Ally at the moment but it's not enough to really make a difference. I would put your money in an online savings account and accrue just a little bit of interest. That way you're not losing as much money. It's not anything serious but it'll help you bleed less money from the interest on your loans.
Online is great. Stay far away from BOA. Awful service and lots of hidden fees. Can't vouch for Chase or TD. If you want a brick and mortar bank, I would go with a local/regional bank or credit union as opposed to one of the megabanks.
I've had extremely great service from BofA. If you keep enough money with them to get preferred status, they treat you like royalty and there aren't any fees. ML accounts count toward preferred status.
I really like Charles Schwab. Opened an account years ago just for their no-fee international ATM withdrawal (saved me a lot of money over the years of travel) and have since moved all banking business there. If you have a ton of cash sitting in savings, I'd go with Ally for their rates.