This sea separates Britannia from the rest of the continent. On its narrowest point it is only 32 km wide (20 miles). At that point it is actually called the Strait of Oostende (aka the Strait of Dover, for our english friends, or the Strait of Calais for our french friends). At a bright day, you can see the white cliffs of Dover from Belgian soil, without binoculars or other artificial means.
As a consequence of its geographical features, the Channel is (and has been for many centuries) among the busiest waterways of Europe. It has seen them all, and plenty of them: trade ships, mines, pirates, privateers, Santaclaus' steamer, smugglers, U-boats, and fleets transporting hence and forth, or simply battling each other. The allies crossed here in WW II. William the Conqueror did the obverse in 1066. And so did William and Mary around 1695, after embarking their army in on the Flemish shores. And it was in this narrow sea, that the Spanish Armada was taken apart, while waiting to board a delayed army in front of the port of Calais (the french port closest to England).
On the diplomacy board it borders 8 other provinces; three of which sea provinces. It separates an english sc (London) from a french sc (Brest) and is therefore a sensitive spot in the opening strategies of both countries (see also the contribution of Blaine Bisanti). There is a third sc that borders The Channel: Belgium; one of the hottest items between England, France, and Germany in the early phase of the game.
As far as I know, the province does not have a prominent role in stalemate lines. And probably rightfully so. If you have leverage on the Channel, you have plenty of options with respect to its surrounding provinces. The Channel means opportunity and action, not exactly the best feature in designing stalemate lines.
From what I've seen, the english channel is a key to british expansion. British control of both the channel and the north sea are very dangerous for other players. Depending on the british outlook, they can expand south or east from these two locations.
Playing France against britain, you would want to tie up the channel with bounces, or if possible, occupy it with a French fleet. Many british-french alliances have come to a bitter end through the british channel. As a safety measure, an attempt by a french fleet from brest to the english channel is always a smart play in the first move, but is often times seen as a threat through british eyes.
A brest-to-english channel move will also distract London away from Belgium and holland, which may allow Paris to move up for the chance of a year-one build. This all depends on how keen france is to ally with germany against britain.
When you isolate britain by clogging up the waterways to the south, you will have a chance for personal gain in both france and germany, and you remove a potentially dangerous power from crossing over to mainland europe early on.