Hi there. Simply putting a USB wireless adapter into the Sony won't work as you can't configure it from the player and it's not designed to work like that.
The solution however is relatively simple.
Buy an additional wireless router that has the facility to be an AP (access point). After you configure this with any PC or laptop by entering your WEP codes, etc, you just place this wireless access point behind your stereo rack and connect the Sony to it with a short network cable.
The player will then be able to communicate with your main wireless router.
With the addition of a cheap multipoint switcher, you are able to connect several units to the same access point, such as an Xbox 360, media PC, network enabled TV sets, etc.
A wireless access point such as this should cost around the £30 price level.
I personally use the CNET CWA-854 but D-Link make some good ones as well.
A lot of people are having trouble with this BD Live business, probably why the newer Sony BD players and all-in-one units have WiFi already built in.
Also, don't forget to purchase the extra 2 Gbyte memory expansion that goes into the back of your BD-350 to enjoy the full BD Live experience.
I've found the best one to be the Sony MicroVault stick.