I don't know of a movie that employs Intelligent Design theory as part of its plot. But there was a "Star Trek: The Next Generation" episode that did: "The Chase":
"After studying the ambiguous number blocks for hours, the discovery is made that these fragments are compatible DNA strands which have been recovered from different worlds all over the galaxy. The crew eventually believe that they have discovered an embedded genetic pattern that is constant throughout many different species, and it is speculated that this was left by an early race that pre-dates all other known civilizations. This would ultimately explain why so many races are humanoid.
Picard resolves that the answer to the 'puzzle' will be revealed when the remaining DNA samples are obtained, and so the Enterprise travels to a remote, uninhabited planet that Galen had mentioned was his next destination. They encounter Klingon and Cardassian ships that appear to be on the same trail as themselves. These two groups believe, respectively, that the puzzle will yield the design of a formidable weapon and the secret of an unlimited power source.
The Enterprise hosts representatives from the Cardassians and the Klingons, and they all agree to combine the DNA samples that they have found so far, since all three parties have pieces of the puzzle that the others cannot find. Using the shared information, they determine a pattern in how several planets were aligned millions of years ago and extrapolate the position of a final planet....
They locate the final DNA fragment, which completes and runs the program. The program reconfigures the tricorder's emitter to project a holographic message. The recorded image of an alien humanoid (Salome Jens) is projected to the assembled company, and it explains that its race is responsible for the presence of life in the Alpha Quadrant.
When the alien race first explored the Alpha Quadrant there had been no humanoid-based life other than themselves, and so they seeded various planets with their DNA to create a legacy of their existence after they had gone. The alien ends its message by saying that it hopes that the knowledge of a common origin will help produce peace."
Here, the ancestral alien race has also bio-engineered the DNA, so that various parts of it are spread throughout the Alpha Quadrant, as pieces of a puzzle, which when finally put together lead to an answer of their origin. I'm willing to bet that most ID critics would admit that this would be a case of Intelligent Design.
However, the Star Trek episode doesn't explain the ultimate origin of DNA. ID proponents argue that it also is best explained as the product of a mind. The jury is still out on this more fundamental question.