Much of the Constitution was deliberately written in broad terms, for reasons of futureproofing.
Certainly, not even the smartest attendee of the Constitutional Convention could have ever foreseen DNA tests or GPS tracking or electronic snooping. It wasn't even something they could have conceivably imagined at the time. But the Fourth Amendment is clear on the matter nonetheless:
The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated...
DNA is, perhaps, one of the most comprehensive pieces of information contained in one's body, one's "person". It can reveal everything from family lineage (ancestry, siblings, and descendants), to congenital diseases or conditions, to the color of one's eyes. It is not equivalent to a fingerprint, which in itself tells you next to nothing about the owner of that finger other than as an identification. The Fourth Amendment is clearly intended to restrict violations of one's person in that way without justifiable cause, even if the particular method of violation is one the Founders would never have conceived of.