Its fine. When you're on the browser, at the screen where you need to log into your account, up near the signal bars there is a lock. Usually if the website is safe the lock is closed & you have nothing to worry about.
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yea i always go into my banks online log in to check balances and stuff. most banking websites offer secure log-ins anyways so im pretty sure its just as secure, if not more, than logging in at home.