What this means is that your bank's server supports TLS-1.0, not 1.1 or 1.2. Versions 1.1 and 1.2 adds some extra features, but that does not mean that TLS-1.0 is weak. In fact the only relevant security feature added by TLS-1.1 over 1.0 is the per-record random IV. This feature inherently protects against some chosen-plaintext attacks, namely the so-called "BEAST attack". However, this attack can also be defeated with SSL-3.0 and TLS-1.0 if the client implements a workaround known as the 1/n-1 split, and Firefox implements it. Also, BEAST can be leveraged only by exploiting a Same-Origin-Policy hole, and no such hole is currently known in current browsers (the two which were used in BEAST demonstrations have been fixed since that time).
For these two reasons, you should not worry about TLS-1.0. In very general terms, it is best when people deploy new protocol versions, so that the new features percolate worldwide; but there is no urgency.
When your bank account gets plundered, it won't be through a TLS protocol hole. Malware installed on your machine is a much bigger threat, indeed.