This editorial account from 1889 has some good information for you:

Aside from its merits, the fate of the St. Thomas treaty was of great advantage, in that it established against a monstrous assumption of one-man power the prerogative of the Senate to act on all treaties with absolute freedom of judgment, unhampered by executive initiation and pledges.

The purchase of Alaska from Russia is not analogous to the attempt to acquire St. Thomas. In the one case the territory was continental, in the other extra-continental and insular; in the one case lying to the north, and in the other tropical; the one bringing wealth in fisheries and furs and fair climactic conditions, while the other was without resources actual or undeveloped, and even subject to derangements of nature unparalleled within the same limited space; the one checkmating the colonial empire of Great Britain in the northwest, and opening the way to the dominion of the continent which has been the thought of far-seeing statesmen like Sumner and Cobden, while the other was to bring us two worthless islands the size of a county, two of the thousand in the Caribbean Sea, with a waste of money in peace and complications in war. Alaska exceeded half a million square miles and the price was $7,200,000, while the bargain with Denmark called for $7,500,000 for a meager area of only seventy-five square miles.

Apparently Seward was acting a little bit too much like Teddy Roosevelt a little too early in American history; and St. Thomas got whomped by an earthquake, tidal wave, and hurricane in quick succession while the treaty was being debated. Not enough gain for the liability.