If you go to Devil's Tower, you will almost always see people climbing up it. Any visitor may climb Devil's Tower! Be aware that they do NOT allow climbing in June because of the Native Americans, so if you plan on climbing, don't come in June. Register with the park to ensure that you will be able to climb when you arrive.
You can also climb up on the rubble (like the people in the picture) for a little fun. Just be careful, since there is a possibility that you may get hurt.
The website provided contains a lot of useful information about climbing Devil's Tower.
Or don't. It looks terrifying to me. But for some, lots actually, it is an adrenaline rush. Our hosts at the Devil's Tower Lodge are avid climbers. They will take you up there and teach you to climb. pretty cool eh? Frank, the big kahuna at the lodge once climbed the thing 17 times in one day. If I ever was to conquer this big rock, he would be the guy I would want around.
In the picture there is a teeny, tiny little speck of yellow on the mountain, that's a climber! That should give you a sense of scale!
Equipment: The Devil's tower lodge folk will help you get whatever you need