I do not hesitate to access my bank via Macdonalds wifi. I have been watching the news and have never heard of anyone losing money through hacking into online banking except the guy who sent money to himself and claimed a thief did it. You are only expected to take reasonable precautions with your password and check your balance at least once a month. The banks are confident about their https - which is far better than credit card or debit card security. I use one bank that says in its terms that the bank is responsible for fraud losses and another that does not. I happen to know the manager of the latter and she says that bank also covers any losses that might occur.
If it worries you, speak to someone at the financial institution and arrange to limit access to what you plan to do online. Strike up an acquaintance so you can establish your identity when phoning the place.