It is not wrong to strum from the wrist, here is a video showing proper strum technique.
http://www.youtube.com/watch?v=sqe9-0xjcK4
With this technique you can rest your arm lightly on the body of the guitar. This will help your hand keep a consistent distance from the strings. You really want to keep your hand quite close to the strings.
It is always good to start with all downstrums. It is common for the upstroke to be a little more difficult, but it comes with practice. It helps to use a soft flexible pick at first. As your upstroke technique improves you can go to harder picks.
The biggest secret to strumming is to relax and play with feeling. Just as a drummer puts accents on certain beats, when you strum the guitar it is the same. It is not a repetitive monotonous thing.
Here is another good video of strumming, note how relaxed the player is and how he plays with accents and feeling.
http://www.youtube.com/watch?v=lMA_lE9fsmM
Just stay at it and it will come to you.
If you know something better than Rock and Roll, I'd like to hear it - Jerry Lee Lewis
