In response Mr. Biden’s deputy campaign manager, Kate Bedingfield, issued a strongly worded statement, saying Mr. Trump’s presidency was “melting down.”

“Donald Trump is terrified of Joe Biden because the Vice President would beat him like a drum,” she said.

No evidence has emerged that Mr. Biden intentionally acted to aid his son in his work in Ukraine.

Mr. Biden’s return to the campaign trail on Friday represented the start of a critical and unpredictable phase of his presidential bid: He is now not only running to lead the Democrats against Mr. Trump in 2020, but he is also grappling with a national spotlight on his family, and how to handle questions — and any potential political fallout — associated with his role in the Ukraine story line that House Democrats are now investigating.

Mr. Trump has been regularly attacking Mr. Biden and his son over their dealings with Ukraine, often using innuendo and falsehoods. The president has also delivered deeply personal insults about Hunter Biden, seemingly trying to bait the elder Mr. Biden into retaliating.

How Mr. Biden handles Mr. Trump’s constant broadsides — as well as questions from reporters and voters about Ukraine, impeachment and his son’s business history — will be a significant test for the candidate who has topped polls for much of the primary race but has seen his lead dip recently.

“You have to be strategic and thoughtful about how you respond and when you respond and the nature of your response,” said the former Iowa governor Tom Vilsack, who attended a recent Biden fund-raiser but has not endorsed any candidate. “He has focused as much on the damage to the country, the concerns he has about misuse of power, as he has on the personal side of this.”