Check bank’s reputation
Does this bank have a reputation of having a bad customer service? Does a bank have numerous hidden fees, that you have to pay for things you even didn’t know about? Do they treat their customer as necessary evil? Probably not much will change in a few years after you get an account there.
Check when your bank was estabilished
As long as money exist, bankers were and will be present. And as long as there are bankers, there will be banks. Some of them are hore for the past 200 years, some of them were created in past 10 years.
Bank with a long history will probably still be around in a few years, not gone with your savings or investments. However those created later may have more convenient and appealing offers or interest rates.
Check bank’s family, sisters and brothers
If a bank is a younger sister (or brother) of a bigger institution that is good. First of all, they may be run by their management. And those are people with experience, less likely to make mistakes newbies would make.
Check the interest rate
Due to our current (in some cases past) economic crysis banks have lowered their interest rates to zero or values such 0.1% (tenth of percent). Search for a bank offering as big interest rate as possible (of course it still have to be a good bank). Even 0.1% is better than nothing. When you put your money in it would not devalue so much due to inflation. It is still an investment.