Support for Russian Prime Minister Vladimir Putin has dropped to the lowest level of his national political career, after parliamentary elections that were harshly criticized by opposition politicians and large outdoor protests nationwide, according to the first polls published since the vote.

The approval rating for Mr. Putin—who has held office as president and prime minister for 12 years, and is seeking the presidency in March—tumbled 10 percentage points to 51% in a survey conducted among 1,600 people from across Russia...