Ms. Gabbard has recently risen as high as 6 percent in polls of likely voters in the New Hampshire Democratic primary. But that support has largely come from moderate and conservative voters, and from independents — not the liberals and partisan Democrats that compose the party’s base. In a Quinnipiac University poll this month, she was the choice of 6 percent of likely Democratic primary voters in the state, but only 2 percent of liberals. Among moderates and conservatives she pulled 9 percent support, and she was at 10 percent among political independents.

One analysis of Ms. Gabbard’s base found that her supporters are overwhelmingly male and more likely to fall outside of traditional Democratic circles, such as those who have backed Mr. Trump in 2016, hold conservative positions or identify as Republicans — particularly in New Hampshire.

Both polls qualifying her for the November debate surveyed voters in the state. Capturing at least 4 percent in one more survey will put her on the stage in December.

The likelihood of her remaining on the debate stage until next year has raised concerns among some of her rivals, who view her as an unpredictable adversary who can deliver withering attacks.

People close to the campaign expect her to try to target Senator Elizabeth Warren of Massachusetts in this week’s debate. Last month, her efforts to quiz Ms. Warren on her qualifications to be commander-in-chief were aborted after the moderator moved to a commercial break. Ms. Gabbard’s campaign later claimed that she was “cut off” by CNN, which was televising the debate, to “protect Ms. Warren.”