Denmark started out with St. Thomas and after 10 years illegal expanded to St.John whom the English claimed, because of the good relationship the English didn't throw the Danes of Island Denmark could maybe if they was careful spread to Tortola which only was settled by English rather late. A good time would be under the War of Spanish Succesion, especially if we avoid the Great Northen War*, of course in that case Denmark may join the anti-French alliance and get rewarded with some Spanish and French territories, maybe Puerto Rico (and the islands arond it USA gained in Spanish-American War)and St. Croix (which Denmark received later in OTL against allowing a French fleet into the Baltic in one of the Polish Succesions wars) they had low population and lay close to St. Thomas. Voila we have a strong Danish base in the West Indies.
What would happen likely Denmark would mass import slaves, the problem with Puerto Rico are that it has room for them to flee into the jungles and highlands, and it's less good for sugar production than the smaller islands, so likely we would see a greater White emigrations to Puerto Rico while the Sugar productions focus more on the smaller island where fleeing are harder for the slaves. So by 1800 Puerto Rico are 50/50 White/Black (as in OTL) while the other island are split 10/90 White/Black (here's Danish West Indies the example).
For Denmark it will mean a stronger economy in the 18th with a larger productions of sugar. That will to a large degree be invested in the army and navy, in oTL the army at it highest was at 5% of the population and the only reason it wasn't larger was because the economy was unable to fund more. We know that the biggest army per capita in the periode was Hesse-Kassel with 7%, so that would be the potential top. While the navy was always kept a little stronger than the Swedish.
Beside that the Danish crown tended to fund industrial and agricultural experiment when it could afford it. One example was the Rhinish peasant imported to the Jutish heath, further settlement was given up because that the state got into economical trouble because of a conflict with Russia. But they did introduce the potato to Denmark, through it took 60 years to become common used.
*The POD could be that Denmark gained Gottorp in the Scanian War, in OTL the Danish invasion of Gottorp was how the Great Northen War started.