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.