I'd contact bank and turn off internet access to your account. I might even change banks. I know this would be a pain, but do not know what to do.
Read through all the stuff below to increase security, but forgoing internet access to your account is the only way to be sure.
I do not use Internet banking. Everyone is being hacked these days. I have remote access to one of my bank accounts, but cannot do much but look at my account. I have a discover card and Discover lets me take out $80 in cash for free when I am at the grocery store. I use a credit card for every payment that I can.
As someone said earlier, you could reload the software on your iPad.
You could try a different browser, like Chrome. In your situation, I'd recommend using another browser. ios is sandboxed so in theory, no other application should be able to access the data in Safari. That said, there is a lot of malware associated with web sites these days. Perhaps, Safari picked up some junk. Pick another web browser. Use it with only your bank and that's it.
Restore to factory settings by loading the latest version of ios that your machine will support:
Do a full backup of your device. Twice is best -- cloud and computer.
While I recommend backing up, I do not recommend using the backup to reload things on your iPad. Do everything manually.
"You can use iTunes to restore your iPhone, iPad, or iPod touch to factory settings and the latest version of iOS."
http://support.apple.com/kb/HT1414?viewlocale=en_US&locale=en_US
"You can put your iOS device into recovery mode, then restore it using iTunes.:
http://support.apple.com/en-us/HT201263