NEW YORK—Some of Sen. Elizabeth Warren’s supporters are starting to worry about her ability to appeal to a broad range of voters outside of liberal hubs, with less than a month to go before the Iowa caucuses.

Ellyn Rothstein of Brooklyn said she has been leaning toward supporting the Massachusetts Democrat in New York’s primary in late April. But she said Ms. Warren’s recent dip in national and Iowa polling—which came as the lawmaker saw a slight decline in fundraising in the last quarter—has made her concerned about the Democrat’s...