Baur was born at Harlingen in 1767, the son of the portrait painter Hendricus Antonius Baur (1736–1817). He painted landscapes, views of cities, and moonlight and winter scenes; he was particularly successful in marine subjects. He is considered one of the best of the later Dutch marine painters. He died at Harlingen in 1820.
Baur became a correspondent of the Royal Institute, predecessor to the Royal Netherlands Academy of Arts and Sciences in 1809.