All current codes require ducts in new construction to be sealed. When reusing existing ducts, that might be a bit of a grey area, but some inspectors will consider it to be "disturbing existing construction" which requires bringing it up to current code... which means sealing the old ducts where possible.

Note that I didn't say "where easy" but "where possible." That means that if the duct is accessible, even in a hard-to-reach (but still reachable) crawl space, it must be sealed. If it's enclosed in a building chase (such as a drywalled box from the basement to the attic) then that's not accessible and sealant wouldn't be required -- unless you opened up that chase during other remodeling.

In a previous Ask TOH episode (I'm too lazy to look it up for you) they demonstrated a method of sealing ducts from the inside using a vaporous sealant blown through the ductwork. This will not only seal the accessible areas, but the inaccessible areas, too.

P.S. -- the rest of your basement dirt floor should have a vapor barrier applied.