What benefit is there to shutting down the banks in Greece? It seems that this would do more harm than good. Won't this impede economic growth by not allowing companies and startups access to capital? This, I believe, would cause the crisis to last longer or get worse.
The deposits that people have in the banks is a lot more than the money that the banks actually have. This is due to fractional reserve banking.
As a consequence, if everybody tries to take out their money, the banks will run out of cash: the banks themselves will be unable to meet their financial obligations, and go bankrupt.
When the bank goes bankrupt, people lose their deposits. Normally the government guarantees the deposits, but in the case of Greece the government is running out of money as well, so it will not be able to meet this guarantee. Usually a government can print money to meet the demand anyway, but Greece is in the Eurozone so it cannot do this (and the European Central Bank is unwilling).
That's why there are closed banks and capital controls (i.e. at most €60/day withdrawal and other limits). It's intended to be temporary. Right now there is a lot of unclarity what will happen. When there is more clarity the banks will surely open again.