I can't speak to restrictions on pleasure boating in the USA, but in Britain, civilian sailing in coastal waters was generally restricted. However, these restrictions changed over time, and depended on which part of the country you were in.

On the outbreak of war, the British government published several Notices to Mariners. These prevented civilian boating activity in several areas, mainly those around naval bases. For example, the waters of the Solent, between the Needles (the westernmost point of the Isle of Wight) in the west and Chichester Harbour in the east, were designated as the Portsmouth Defence Zone. In this zone, civilian sailing was allowed on the inshore rivers and creeks, but only in daylight hours. Those wishing to enter coastal waters had to have a permit, and again could only operate in daylight. However, entry to the waters immediately around the naval base at Portsmouth was much more heavily controlled. In April 1940, a new order, Navigation Order No. 5, was introduced. This restricted entry to Britain's tidal waters to rowing boats, fishing boats and merchant vessels, with civilian sailors requiring a special permit from the local naval authorities. However, these permits were generally freely given, and geographic restrictions began to relax.

This meant that during the early part of the war, sailing survived reasonably well. Many in Britain's boating community looked forward to the 1940 sailing season, and a considerable number planned race meets for that summer. A survey of 73 yachting clubs, carried out by the magazine Yachting World in May 1940, received 56 responses from clubs planning to set up races, and just three from clubs that weren't, with one of those being the Royal Navy's own Royal Naval Sailing Association. Unfortunately, the Fall of France brought a swift end to these plans. With the threat of German invasion looming, new orders placed strict restrictions on coastal sailors. On the 9th July, the Vessels (Immobilisation) Order came into force. This aimed to ensure that any German invaders could not use captured British vessels to move supplies and troops around coastal waters. In the most threatened areas, the southern and eastern coasts of England, it required all vessels on tidal waters or rivers to be beached or removed from the water (and kept as far away from it as possible), and immobilised by removing key components or by holing the craft. In the rest of the country, boats could be kept on the water, but had to be immobilised if left unattended. In August, a further order placed even stricter restrictions on sailors.

These restrictions were enforced with fines or stricter penalties, with a number of violaters being threatened with having their vessels sunk or destroyed. However, as the war continued and it became clearer that no invasion was imminent, the restrictions began to lift. In May 1943, the restrictions were lifted on inland waters across much of the UK, with the exception of the south coast and East Anglia. In September 1944, restrictions were further relaxed, with immobilisation orders being removed on all inland waterways and on coastal waters outside much of the south coast. Over the course of 1945, the orders continued to relax,and were removed completely following the German surrender. While the restrictions were lifted across much of the country, they remained in place around key naval bases, such as Portsmouth, Greenock (on the Clyde) or Liverpool, until the end of the war.

Beyond the legal obstructions to sailing, there were a number of other disruptions to leisure boating. The Royal Navy established boom defences around a considerable number of British ports, making it harder if not impossible for boaters to leave those ports. Boaters and their vessels were frequently called up for war service. The Royal Navy had, in 1936, established the Royal Naval Volunteer (Supplementary) Reserve, a list of yachtsmen who would join the service as officers in time of war. For those who had not joined the RNVSR, merely holding a sailing qualification made it more likely that one would be called up by the RN over the course of the war. Royal Navy Volunteer Reserve commissions were available to any experienced yachtsman up to the age of 45. A number of others with power-boat experience were called up by the RAF, to man the Air-Sea Rescue service. There were also a considerable number of voluntary organisations which those who were not eligible to be called up could join. These services included river patrols in the Thames, Clyde and Trent, which patrolled these waters to mark German mines, watch for sabotage attempts against bridges and locks, and provide a nautical counterpart to the Home Guard. Boats were also eligible to be called up. In 1939, the Admiralty set up the Small Vessels Pool, which listed boats that could be called up for use as ferries, crew transports, tenders and patrol craft. The most famous outing for the Small Vessels Pool was Dunkirk, when the craft were called up to supplement the Navy's own boats to evacuate men from the beaches. The craft of the Small Vessels Pool continued in less cinematic roles for the remainder of the war. The Admiralty also used the Small Vessels Pool to call for yachtsmen to ferry newly built craft (torpedo boats, landing craft and the like) around the country (or even across the Atlantic) from the builders yards to where they were needed. Boats were also called up by local organisations, to block waterways or, in one case, to hinder possible seaplane landings. Yacht builders were more concerned with building coastal craft than with building civilian yachts or powerboats.

Despite these difficulties, yachting and boating did continue throughout the war. Many inland waterways were still available for use. Rivers , reservoirs and lakes were used frequently. West Kirby Sailing Club (my current club), despite being based on the tightly controlled waters around the Wirral Peninsula, was able to use the local marine lake. A number of naval officers were also able to indulge their passion, sailing in the waters around their bases, while a Naval Sailing Club was set up on the River Bure, for the benefit of sailors on leave.

Sources:

Troubled Waters: Leisure Boating and the Second World War, Nigel Sharp, Amberley Publishing, 2015

Churchill's Navy: The Ships, Men and Organisation 1939-1945, Brian Lavery, Conway, 2006