Did they support the North? Send troops or aid? Sanction one side or the other?
Interestingly enough, the US Civil War is often cited by historians as one of the major reasons for Canadian Confederation in 1867.
Canada was not a single country at the time, but rather was still a collection of British provinces. Though by this time they were all essentially self governing, they had no control of their foreign affairs, which were handled by the British Parliament.
That being said, Britain did take a side. As you may or may not be aware, the British supported the Confederacy during the Civil War. The British textile industry relied on the cheap cotton they imported from the American South, and this cheap cotton was made possible by the slave economy. During the Civil War, tensions between the US and the UK reached their most dangerous levels since the War of 1812. In 1861, Union ships boarded the Trent, a British merchant vessel en route to London that was carrying two Confederate ambassadors. Even worse, the CSS Alabama was built by the British for the Confederacy, and it wreaked havoc on Northern shipping until Union ships chased it across the pond and sank it off the coast of Normandy.
Thus, when the Union emerged victorious in 1865 Britain's North American colonies were faced with southern neighbour that was fully mobilized and looking for a way to get back at the British. They did so in two major ways. First, at the end of the Civil War the US government refused to renew the reciprocity (free trade) agreement that they had signed with Britain's North American colonies. Second, the Fenians -- a group of radical Irish nationalists who sough to free Ireland from the british by taking Canada hostage -- launched a series of raids from the US into Canada in 1866, and Washington turned a blind eye.
The provinces of Canada (which existed from 1840-1867 as a unified colony of what is now Ontario and Quebec), Nova Scotia, and New Brunswick were negotiating the terms of Confederation at the time, and realistic possibility of another war with the United States lent a new urgency to their discussions. The Quebec Conference (1864) hammered out most of the principles that would be put into the British North America Act (1867; Canada's constitution), and it was up to the representatives of the provinces to sell the idea of union to their voters. The provinces of British North America had relied heavily on trade with the US, and in fact they did little trading with each other. Furthermore, raids into Ontario and New Brunswick by the Irish nationalists caused quite a stir. A political union between the provinces of British North America offered a chance at internal trade to replace what had been lost with the US, as well as a cohesive defensive strategy. The Quebec Resolutions were accepted, in some cases begrudgingly, by the provinces and on July 1, 1867 the British Parliament passed the BNA Act into law. Ontario, Quebec, New Brunswick, and Nova Scotia became provinces of the dominion of Canada.
The Treaty of Washington (1871) saw the British repay the Americans for the losses caused by the CSS Alabama.