I would consider the very real threat against the other continental Monarchies and the British system itself as the overarching distrust of Britain and their financial and military support of both the 1st coalition and all of the coalitions to follow.

Support for the French Revolution in Britain challenged the entrenched parliamentary system in Britain. William Pitt the Younger strongly opposed the "radical" liberal reformers in Britain. Keep in mind that the French Revolution was a revolt against the First and Second estate (Nobles and Clergy), and that Britain was a highly stratified society, susceptible to reform as much as France. Until Napoleon reestablished much of the prestige around hereditary titles, the French prided themselves on simply being "citizens". Although the British parliamentary system may have been more liberal than the French monarchy, the National Assembly, Legislative Assembly and National Convention were far more radical than anything seen in Britain.

Another key issue was overseas colonies. French control of St. Domingue generated an obscene amount of income for the nation. However the Girondins pushed to abolish slavery, and abolishing slavery in St. Domingue would be (in the eyes of the British) disastrous for British slave colonies in the Caribbean. Taking St. Domingue from France would not only choke off most of their colonial income, but would help protect British colonies from slave revolt.

This also folds into another issue for Britain. The British were far more technologically and economically advanced than any of the continental powers, but their strength was in sea power and economic influence rather than strictly military dominance. Britain maintained small standing armies. While they could mobilize their population over time and eventually win wars (as evidenced by their grinding campaign of attrition in Portugal and Spain in the waning years of the Napoleonic era, as well as most of the wars in which they fought in the next two centuries), it was in their direct benefit to keep all of the continental powers in relative balance. The French revolutionaries sought to extend their ideology beyond the "natural boundaries" of France, spreading into Prussia and Austria and beyond. While this could be achieved militarily (Austrian controlled Belgium in 1792/93), it could also be achieved ideologically, or so the revolutionaries reasoned. The spark of democracy in France would light the revolutionary beacons in Prussia, Austria, Italy, Holland, Russia, Britain, etc. Wrapping back around to the first paragraph, this posed a direct threat to Britain and their gentrified system. Napoleon actually pushed to isolate Britain economically via the "Continental System" as his power grew. One of the conditions of the Treaty of Tilsit with Russia after the Prussian campaign was that Alexander and Russia join the Continental system.

I don't think any of these individual reasons really tell the whole story. You would have to look back to the historical rivalry between Britain and France to find a more complete picture. The French monarchy was more or less bankrupted by its support for the colonies in the American Revolution so soon after the Seven Years War. Louis XVI was not exactly a friend of Britain, but maintaining the status quo mattered to the entrenched powers in Britain. You also have to consider the marriage between Louis XVI and Marie Antoinette and the destabilizing force of the Revolution in relation to Austria and the HRE. There are so many factors that it's hard to comment on them all, as they could all be described independently but fit into a larger framework.

Sources:

William Pitt the Younger, by Eric Evans Citizens, by Simon Schama The Coming of the French Revolution, by Georges Lefebvre