There are products out there on the market that are specifically designed for this purpose- things like this DVD guard or this TV guard- both sell for around $10. My version, though, only cost $2.66, and so far, it's working great for us!
I went to WalMart and I purchased two 5x7 clear plastic picture frames. I slid the "stand" part of the frame up under the edge of the DVD player, and used the frame part to cover the on/off buttons. I did the same with the other frame, using it to cover the area where the play, fast forward, and other buttons are located. It took less than a minute, and so far, it has kept busy one year-old fingers away from the buttons. (I'm sure he'll eventually figure out how to remove the frames, but, hey, it works for now!)
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