This frost affected Babaco will come back next spring as good as gold. Just leave it alone until spring, then trim off old foliage.
When confronted with the soggy brown / black disaster of frost affected foliage your first instinct is to grab the secateurs and prune away the evidence, but just take a minute to consider. If it is a potted plant that you plan to move indoors for future protection, then by all means prune away the unsightly mess and bring inside. Similarly if it is a soft stemmed plant, it may rot or get fungal issues, so prune it back (if the plant has been cut right down and is remaining outside, put straw or mulch over the soil to protect from further frosts). However, if the plant is a woody or semi woody plant, and there is potential for further frosts, wait until the last frost is over and prune in spring. The old dead material will help protect the undamaged stems. Prune dead material away in spring, and the plant will generally recover.
Preventing frost damage
This is easier said than done sometimes, but the first step is to keep an eye on the overnight weather minimums via the Bureau of Meteorology. If it looks like the chance of a frost, then I prefer to use a double covering, the top layer catches the frost and the bottom layer protects the plant from the frozen top layer. Also make sure the soil is damp not dry before the frost occurs. If is in an option, bring any susceptible pot plants inside.
Frost and succulents
Many succulents will survive a short mild overnight frost (-1C), they will look damaged (like the pictured Crassulas) and need to be cut back, but they will live. Some succulents, like Sedum reflexum, Sedum sieboldii, Sedum spurium ‘Voodoo’ and ‘Tricolor’ and most of the Sempervivums will tolerate prolonged freezing conditions and look fabulous. If you are in alpine conditions – pick these succulents.