Occasionally when I observe people playing games with a joystick (especially in an arcade), I'll see them grab the joystick full-on with their hand and hang onto it like it's the only thing keeping them from falling off a cliff.

Which works, yeah, but it definitely hinders your experience by requiring that you use your entire arm to play which results in:
The solution?
Grip the joystick lightly between the thumb and the index finger, using the middle finger as well if that's more comfortable for you

You get more accurate control and don't use so much energy, which leads to longer playing sessions. It's totally win-win!