I was about to use an online MD5 checksum calculator when I realized that my result could be added to a database. This is significant because the string I was going to type in was my bank password. Could online hash calculators be dangerous?
Yes, submitting your password to any website is dangerous.
There's nothing special about online hash calculators, think of it as no different from typing it in to Google or posting it on StackExchange. Not only could that website be storing the entered password but if you aren't connected over HTTPS then anyone could sniff that password.
If you want to calculate the MD5 of your password (though I'm not quite clear why you would?) do it locally - there's plenty of software a Google away to help you.
Potentially yes, the hash could indeed be stored in a database for malicious use.
Never input your password into a third party website unless you trust the provider of the service AND the service is using some sort of authentiation/encryption scheme(SSL, HTTPS).
You could easily generate your own hashes on your own system, using MD5 libraries of various programming languages.
Of course anything can be considered dangerous - you certainly should not be putting any sensitive information into a system when you are not sure what's being done with it. Also a lot of these online calculators don't use https so your password would be sent in the clear anyway, so thats bad before getting started with what they are doing with it.
If you are wanting to calculate hashes you can do this locally on most operating systems particularly with something as simple as MD5, where there is support for it in almost all programming / scripting languages. So it would be best to do it locally unless the hash you're wanting to calculate is not sensitive anyway.
For example you could calculate it locally using python in 2 lines
import hashlib print hashlib.md5("whatever your string is").hexdigest()