According to the Bulletin of the New York Public Library (note- Hathiturst links don't work for some countries) the cartoon in question was in response to the escalating naval war in the Baltic.

The British attack on the Danish fleet and the bombardment of Copenhagen in 1807 had prompted a Danish reaction to the British incursions. The shock of the bombardment meant that Danish authorities were more receptive to Napoleon's demands that they be more proactive in enforcing the Continental System of blockade. Anti-smuggling initiatives would seize ships suspected of violating the Continental System, which in turn would be used to fill the ranks of the depleted Danish fleet.

American merchantmen were collateral damage in this Baltic maritime war. The Americans ships were not innocents in this matter as American merchant ships were active in smuggling operations. The latter half of 1809 witnessed a slew of seizures by the Danes of American ships on the pretext that they had false papers and were carrying British contraband.

Many merchants appealed to both the US consul in the northern Germany as well as to the US Secretary of State and President. As the cartoon suggests, there was a sense that diplomatic protests matched with a Yankee determination to resort to force if necessary was an adequate solution to this problem, but these were outrages the young US should not forget.

Tom Holberg at Napoleon-series.org has a good digital piece, "Danish Privateering, 1807-1811" which has some of the background context as well as digital transcriptions of the diplomatic protests from American officials and merchants to Denmark's actions.