Dynamite is, basically, filler material soaked in nitroglycerine then wrapped up in sticks. This makes it more stable than nitro, but as time passes the nitro sweats out of the dynamite making the sticks less and less stable.
The sticks in the movie had been shown to be old and sweating, so, theoretically, a hard knock could have set them off. Or an arrow.
More brains for the Brain Machine!

It has been my experience with dynamite that it deteriorated over time. The sticks just fell apart.

The dynamite was extremely old, and hence extremely unstable. Like Tom Berenger's character said in the movie, dynamite is like wine, it only gets better with age. Now I have never fired an arrow at very old dynamite so I cannot say for certain what would happen, but I suspect the concept portrayed in the movie vis-a-vis detonating old dynamite using an arrow is not unrealistic.