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avast! would protect your computer from any malicious programs or attempts to harm your computer. But it should also be noted that antivirus programs can only protect you online from what’s trying to get inside or outside your computer—all as per your consent. That means anything else that’s already outside your computer (i.e. data sent such as usernames and passwords) would be beyond the scope of security. As it is the case, it is generally suggested to refrain using public wi-fi connections for such transactions as they can be intercepted. Connection through cellular data plan is generally more preferred.
Some good reads:
5 tips to stay safe on public Wi-Fi
Is It Safe To Bank On Public Wi-Fi? How Not To Get Hacked!