I'm trying to decide if I want to give the PSN+ service a try since it offers a bunch of free games. However, the only time I usually use my Vita is times where there is no network access (such as long plane rides). Do I need to maintain a constant internet connection to be able to play the free PSN+ games?
No you don't. The games can be used at any time as long as you still are subscribed to Playstation+. (Though you might have to reconnect briefly if you renew so your Vita knows that you're still active.)
Over the Thanksgiving break I played Gravity Rush, which I'd gotten free with Plus, over a period of five days, without any network connectivity
Note that the games aren't playable after your subscription lapses.
If you try to launch a PSN+ game while on a plane, with "Flight Mode" enabled, some games will not launch, you will just get an error stating "Flight Mode is on."
This unfortunately means you can not play all the downloaded games while on a plane.