It depends, if online only I would honestly just go with a brokerage like others suggested. If you need more traditional banking services then JP Morgan/Chase is the largest and also probably the most stable and reputable, but they do have a minimum balance for their checking accounts. They have an aura that surrounds them due to being very old and surviving the depression. Doesn't mean they would survive a depression today, but it's good advertising at least.

Local credit unions are an option but your mileage will vary. Often membership is not open to the public.