While Rick Smith is absolutely correct that Clinton was not re-elected, there is a common belief that Clinton’s impeachment was damaging to the Republican party, which is likely the basis of this belief you describe. In my answer I will address this.
The fear that Impeachment will hurt the Democratic party comes from the traditional wisdom about the Clinton Impeachment which says that Impeachment hurt the Republican party in the upcoming elections. In reality, though, there is little evidence of such a political cost.
The Clinton Impeachment took place over 1998, the second year of his second term, in the lead up to his second-term midterm elections. The House voted to begin an Impeachment Inquiry against Clinton on October 8, 1998, less than a month before the Midterm elections. In those elections, the Democrats gained 5 seats in the House, while the Senate was unchanged. This was considered a major loss because, to quote wikipedia:
With the Republicans having lost 4 House seats and failing to gain any seats in the Senate, it was the first time since 1934 that the non-presidential party failed to gain congressional seats in a mid-term election. It was also the first time since 1822 that the non-presidential party had failed to gain seats in the mid-term election of a President's second term.
So, from one point of view, the Republican party did pay a cost for their Impeachment. Despite this, however, the Republicans retained their majority in the House and Senate, and therefore didn’t lose any political power.
In the next elections in 2000, despite the Democrats gaining 4 seats in the Senate and 1 in the House, the Republicans won the Presidency and retained a majority in both the House and the Senate (though they lost their Senate majority on June 6, 2001, when Senator Jim Jeffords switched parties).
So, despite losing seats, the Republican party didn’t actually pay any cost for Impeaching Clinton: they didn’t lose control of any part of government, and, in fact, many think it might have helped them take the Presidency in 2000 despite Clinton leaving the presidency with the best Economy in recent memory and a budget surplus.
In a 2019 article in The Atlantic, Ronald Brownstein argues that Impeachment, despite being unpopular and failing to result in a conviction, tarnished Clinton’s reputation and made him significantly less valuable as a surrogate for Vice President Gore, who hoped to succeed him:
in 2000, lingering unease about Clinton’s behavior provided a crucial backdrop for George W. Bush’s winning presidential campaign—particularly his defining promise “to restore honor and dignity” to the Oval Office.
Tad Devine, a senior strategist for Al Gore, the Democratic nominee in 2000, concurs. Bush’s ability to tap the public’s dismay over Clinton’s personal life “more than anything else got in our way in terms of winning the election,” he told me.
After the Senate refused to remove him in early 1999, Clinton’s job approval remained about a buoyant 60 percent through the rest of his presidency. But that strong number coexisted with substantial public disapproval of the personal behavior that the Starr investigation and the impeachment inquiry had highlighted.
Those personal doubts about the outgoing president cast a huge shadow over the election to succeed him—a dynamic usually omitted in the equation portraying Clinton’s impeachment solely as a self-inflicted wound for Republicans. Those doubts led the Gore campaign to conclude they could neither campaign with Clinton nor deploy him extensively as a surrogate.
Both Devine and Dowd noted that Bush’s constant pledge to restore presidential honor and dignity skillfully tapped that unease without fully embracing the divisiveness of impeachment itself.
But Bush carried one-third of the voters who liked Clinton’s performance but disliked him personally. That’s a much higher-than-usual level of defection from the president’s party among voters who approve of his performance, and in 2000, those voters represented about one-fifth of the entire electorate.
Bush reaped another benefit from the impeachment, Dowd believes: high turnout among Republicans frustrated that Clinton remained in office.
Ultimately, Impeachment is so rare that any historical comparison is likely to be tenuous. Yet, while it’s clear that while Impeachment can have risks for the opposition party, the conventional wisdom about the cost the Republican party paid for Impeaching Clinton is not supported by historical evidence.