It is a common factoid that the Dutch were the first to recognise American independence in 1776. This happened at Fort Oranje on St. Eustatius, where the Dutch governor Johannes de Graaff had an eleven-gun salute fired for the American warship Andrew Doria. However this action does not constitute formal recognition of American independence by the Dutch Republic. Whatever his actions, governor De Graaff did not have instructions to act as if the US were an independent nation nor did he have the authority to make such a decision himself.
The British were somewhat angered by De Graaff's actions, but they were even more angry about the Dutch trading weapons and munitions with the Americans. In 1777, the British threatened to proceed with unrestricted boarding and seizure if the Dutch did not stop the flow of weapons and munition. The Dutch responded by preparing the navy and forming convoys. One such convoy bound for the West Indies was forcibly diverted to Plymouth in 1779. During 1780 the British boarded many Dutch vessels in and bound for the West Indies.
It is important to realise that at this time, the Dutch Republic was in decline. The days of the Republic being an economic and military powerhouse were over, though the memory of it remained. The people at the time realised that the situation of the Dutch Republic was no longer as favourable as it used to be. On the European continent it was surrounded by the rising powers Austria (southern Netherlands / now Belgium) and Prussia. Overseas, the Dutch colonial empire and trade routes were vulnerable to Britain's imperial dominance and maritime superiority. Internally, the Dutch ruling class was very unpopular among the urban populace.
The British boardings violated the principle of "free ship, free goods", which had been the basis of Anglo-Dutch relations for over a century, at least since the Treaty of Westminster in 1674. The relative power of Britain and the Dutch Republic had changed since then however. From roughly equal in the 17th century, the British gained power whereas the Dutch lost power. The Dutch saw the American Revolution - and the related end of the British monopoly on trade with America - as an opportunity to try to rebalance the scales. This is why the Dutch did not stop the trade with the Americans via St. Eustatius, going as far as risking war with the British over it.
In the end war did come. British boardings proved the Dutch were continuing their trade, and other revelations made clear that Amsterdam merchants and financiers were heavily involved in assisting the American Revolution. At the end of 1780, the British declared war on the Dutch Republic. They saw the Dutch colonies and shipping as easy pickings which could offset losses suffered at the hands of the Americans and the French. This is when the British captured St. Eustatius (so the capture of St. Eustatius was not, as some American histories may lead you to believe, a response to the eleven-gun salute).
It was in 1782 when the Dutch Republic formally recognised the United States of America. On October 8th 1782 the Dutch States-General (the body in control over Dutch foreign policy and matters of defense) signed a trade agreement with the United States. (I'm not sure 8 Oct 1782 would rank being early to recognise the U.S. or not, I have no idea when other nations recognised the U.S.) Why did the Dutch do it at this point? One reason being of course that both nations were at war Britain. A second reason being the all-important trade. Trade with American colonists kicked off the war with Britain, and the Dutch wouldn't want that (frankly disastrous) war to have been for nothing.
