Seriously, I would change bank if mine forced me to use a Java applet to access my online statement and pay some bills. Your bank should offer the option to use a non-Java version, or ditch Java totally as most online banking requirements are catered for with standard browser HTML, Javascript and SSL - using full blown Java for online banking is insane.
Agreed, wireless banking isn't something I would personally bother with but SSL is sufficiently secure even over a wireless connection to make it a practical proposition.
SSL wired and SSL wireless - there really is no difference! If you access your online banking in the office over a wired connection, how do you know your traffic isn't being sniffed by a nefarious colleague? This is no different to someone grabbing your SSL-encrypted packets out of the air. You're data (and money) is safe.
Right, except your home PC isn't scattering packets to the wind.
Except it really doesn't matter provided the data is encrypted using medium to high grade SSL.
When someone finds a way to decrypt SSL in anything like a reasonable amount of time (ie. near-real time rather than the current many hundreds of years per message, unless you're the NSA) then I wouldn't worry about your online banking, instead worry about the large parts of the worlds economy that will go down the pan as online transactions suddenly become insecure!