How long has it been crispy with no new growth? Cutting any part of it will only add to stress. I suggest you make a firm little nick in the bark halfway up the trunk, and if you see green underneath, it's still alive, so do water from above til it comes out the drain hole (if it isn't already wet from previously), find a cool place to keep it, in bright light, but not necessarily right on a sill. Put a wide tray with high-ish sides underneath, filled with stones and water, with the pot on top, but the water should never touch the pot bottom (or it could rot roots). If in two wks you see no sign of new growth anywhere, chalk things up to beginner's bad luck. Bonsai are not house plants when they're grown as bonsai, and they need certain type of soil mixes, different watering techniques, special lighting and humidity. If you're interested in learning more, don't start by buying anything but books or a membership to a local club (great place to learn!), and go to www.bonsai4me.com for a general idea of what some of it's about. Once you have a better idea of that, you'll be in a better position to decide what to buy. Your experience is another example of why giving bonsai as gifts is a big mistake unless you know the person you're giving to has a clue about how to take care of them or wants to bother! But if you do, it's really rewarding, if results are more long term than other interests.

I have a gardenia (not bonsai) that's in a pot outside.
This winter, almost all of the leaves have become crispy
brown and dead. There is new green growth from the lower
parts of the plant, so I continue to water. But I'll be sure
not to prune until the plant is growing strongly again.

ok, my bonsai is officially still alive, as there is indeed green under the bark on the trunk.

I will keep caring for and watering it, and will put a humidity tray of sorts under it.