Sukhumwit soi 7/1, which is a little dead-end lane lined with bars, many of the bars have free wifi. If you go there in the afternoon when it's quiet, you can use it for the price of a drink. Be prepared to buy a bar-girl a lady drink and politely tell her you won't be needing her company, though.
Landmark Hotel coffee shop has free wifi and the waitresses will even tell you where to sit for the best reception.
In Times Square internet shop at Asoke BTS, I believe you can use the True wifi system for a price. The fastest I've ever had in Thailand. There's also another True centre on Silom.
Personally I wouldn't do any home banking or sensitive emailing on any open wifi system, anywhere.