I used to foster so have a lot of experience with this. Most successful method I've found is being quick. You have one chance.
Time the vet appointment when your cat will be relaxed or sleeping. At 14 I'd expect she does this a lot. Put her in the basket 15-30 minutes before the taxi is due then it's done before they arrive or you have time to cancel the taxi and vets appointment if you can't do it.
Get the basket ready somewhere accessible but not in the same room. Make sure only the top is open. If if doesn't stay open by itself, but needs lifting up, get another one, honestly.
When the basket is ready, go and pick up your cat. If you can just pick her up, great. If not, grab her by the scruff of the neck and under her bum so her weight is supported. Quickly put her in the basket, bum first so she can't stretch her legs out to stop you and quickly zip up.
Throw some Dreamies in by way of apology and remind yourself it's for her own good that she has a regular health check. If she's a shouter, leave her in the basket and out of earshot until the taxi arrives.
People have said you can leave the basket out so she gets used to it and uses it as a bed, and you just shut her in when she has a vet appointment but I've never tried that because cats are psychic, know when they're going to the vets so won't fall for that trick.