You would need a working port to connect to Desktop software, as far as I know. But you can save a contact to the SIM card by opening it, pressing the Menu key and choosing "Copy to SIM Phone Book." I believe you have to do this for each contact individually though, not the whole address book at once.
If you are not seeing the "Copy to SIM" option, you may not have a SIM card in there. And the only way you would use a SIM card is if your Curve is the latest model with global capabilities.
Another option is to use Verizon's Backup Assistant to wirelessly save your contacts. And maybe some of our BlackBerry gurus here know of some other ways to do this wirelessly. Good luck!
What phone are you moving to? If it's another BlackBerry, you can use BB Protect to back things up and then switch devices, and restore your stuff on the new BB using BB Protect - I used it a few months ago when my Bold went swimming, and I borrowed a friend's Curve till mine dried out - BB Protect transferred my contacts back and forth between the two devices quite nicely.
I used to use Verizon's Backup Assistant, haven't used it while I had the BB, but just today, actually, I upgraded to a Droid, and the gal in the store installed and used BUA Plus to get my contacts from the BB to the Droid. It worked well, so I would recommend using it to make a backup from your current Verizon phone - that backup can then be used to restore all your contacts on a new phone.