But as Mr. Sanders fights to close the delegate gap, his comments sent a shudder through party officials aligned with Mrs. Clinton. In her campaign’s planning, April was meant to be a relatively calm month in which to focus on raising money for a general election and honing a message to use against the eventual Republican nominee. Mr. Sanders’s staying power and online fund-raising prowess have instead caused her campaign to spend heavily for advertising in expensive media markets.

But an extended — and increasingly toxic — nomination fight, several advisers said, could deplete Mrs. Clinton of resources and leave scars that make it harder for her to unite his supporters behind her.

“Progressives are going to have to come together in November to defeat whatever crawls out of the G.O.P. circus in Cleveland,” said Jess McIntosh, communications director at Emily’s List, a group that works to elect women who support abortion rights and has endorsed Mrs. Clinton. “There are a few attacks that make it harder to do that, and Bernie Sanders is going there.”

Supporters of Mr. Sanders dismissed the Clinton camp’s reaction to his remarks as hypocrisy and fake outrage. After all, Mrs. Clinton had attacked Barack Obama’s experience during their contentious 2008 primary race. “I have a lifetime of experience that I will bring to the White House,” she said then. “Senator Obama has a speech he gave in 2002.”