Both Hunter Biden, the privileged and troubled son of a president, and Ms. Roberts, the daughter of a rural gun maker, have allies whose actions have made the situation more politicized. There is no evidence the White House is involved in those actions.

Clint Lancaster, Ms. Roberts’s attorney, has represented the Trump campaign. He also called Garrett Ziegler, an activist and former Trump White House aide who has cataloged and published messages from a cache of Hunter Biden’s files that appear to have come from a laptop he left at a repair shop, to serve as an expert witness in the child support case. In the other corner, allies of Democratic groups dedicated to helping the Biden family have disseminated information about Mr. Ziegler and the Roberts family, seeking to highlight their Trump ties.

And then there is President Biden.

His public image is centered around his devotion to his family — including to Hunter, his only surviving son. In strategy meetings in recent years, aides have been told that the Bidens have six, not seven, grandchildren, according to two people familiar with the discussions.

The White House did not respond to questions about the case, in keeping with how officials have answered questions about the Biden family before.

Several of the president’s allies fear that the case could damage his re-election prospects by bringing more attention to a son whom some Democrats see as a liability. Others say the far right has focused on Hunter Biden, a private citizen, but ignored any moral and ethical failings of the former president, Donald J. Trump.