The Strait could and was blockaded with just Gibraltar. Their landward side was cut off by the neutral Spanish. Their seaward approaches were defended by the powerful Force H. The land across the Strait was neutral Spanish Morocco. Gibraltar was fairly secure. Once the Vichy French fleet at Mers-el-Kébir was crippled and later the Italian fleet at Taranto there was little serious threat to Gibraltar. RAF units at Gibraltar could patrol the Strait. The main threat was from German and Italian aircraft.
...but who would they be blockading?
The Allies need the Strait open to shipping to supply troops fighting in North Africa, Malta, Crete, Albania, Greece, and the Middle East. They also needed it to use the Suez Canal to shorten the supply line to the USSR via the Persian Corridor, as well as India, Burma, and the Far East. Otherwise ships had to go all the way around Africa taking far more time, exposing them to attacks, and straining the Allied worldwide refueling system.
In contrast, the Axis did not need the Strait. They had ports all across southern Europe to supply North Africa and a (theoretically) powerful Italian fleet to protect them. West Africa could be supplied from the Atlantic side via French Atlantic ports. The European Axis powers had no major military operations to supply elsewhere that the Strait would be relevant to, most of their fronts were overland with internal lines of communication and supply.
The only exception is moving warships, but even this was of limited use to the Axis. The Italian navy was never designed to leave the Mediterranean to begin with and was content to try and make the Med into an Italian lake. For example, its Littorio-class battleships had a range of about 8000 km while the comparable British King George V-class had a range of almost 30,000 km.
The German navy was primarily interested in the Battle of the Atlantic and other commerce raiding. They were only interested in moving U-Boats into the Mediterranean to disrupt Allied supply convoys, which they could do, with some safety, by remaining deeply submerged and silent and allowing the current to push them through.
After the invasion of the Soviet Union, the Black Sea became a battlefield. Using the Strait would do them no good because the neutral Turks kept the Dardanelles closed to all belligerents. Instead, Axis naval power on the Black Sea was primarily provided by Romania. Some U-boats and small craft were shipped overland, or by river, to Romania, and reassembled for use in the Black Sea.
The question of an Allied blockade at Gibraltar is moot, the Strait was of little value to the Axis. An Axis blockade would harm the Allies greatly, but they could not blockade the Strait without first neutralizing Gibraltar.