Strait of Gibraltar is a body of water between Europe/Spain on the northern side and Africa/Morocco on the southern. In medieval (and modern for that matter) times the key cities were Gibraltar, Tarifa and Algeciras on the Europe side, and Ceuta, Tangier, Arzila and Alcácer Ceguer on the Africa side. Depending on what you consider "taking the straits" there are several scenarios we should analyze.
While we usually think of Reconquista as a slow gradual process, majority of southern Iberian peninsula - Muslim ruled al-Andalus - was taken by Christians in relatively quick succession of events between 1200 and 1250. Almohad dynasty became embroiled in civil war back in Morocco, and the forces stationed in Iberia returned to Africa. This left a huge military and political vacuum in Iberian Muslim states. And they were opposed by resurging Christian forces which were for change cooperative with each other. What followed was the above mentioned massive conquest of Moorish territories. Jamie I of Aragon took Balearic islands and Valencia. Leonese and Castillian forces, first separately and since 1230 united under Ferdinand III took Badajoz, Huelva, Jaen, Cordoba, Murcia and finally Seville in 1248, and reduced to tribute the state of Granada. Portuguese under their king Sancho II also advanced south, (re)taking Alentejo region and then Algarve, finalized with taking of Faro by new king Afonso III in 1249.
During this time, Portugal was in a slightly political mishap. The king Sancho II was in a serious dispute with the clergy and the pope, culminating with pope ordering his deposal in 1245 and replacement with his brother Afonso III, an order carried out seemingly without much consternation by the Portuguese. In a military geostrategical way, Portuguese were physically prevented any expansion east by Leonese conquest of a strategic city of Badajoz in 1227. Badajoz was a key city about which Leonese and Portuguese disputed who had the right to conquer it. (To the point when Portuguese were on the brink of conquering it in 1169, Leonese forces moved to support the Moorish defenders and broke the siege) Following conquest of Badajoz, Leonese proceeded and reached the Atlantic shore at Huelva, effectively creating a land barrier for Portuguese armies to reach Gibraltar. Which anyway they were in no position to do, as they were still embroiled in their fights in south of Portugal and Algarve which didn't finish until 1249. From this point forward, as Gibraltar and Granada was landlocked by Castile-Leon, it was only they who could (and did) engage in fighting over trying to conquer the European side of Gibraltar straits.
For Portuguese the only way they could possibly do so was by sea invasion. And this is an option they seriously considered in 15th century. Which leads us to the interesting element of the story: the Portuguese actually did take the southern/African part of the Gibraltar straits, and remained in control of key points until 17th century. They took Ceuta in 1415, Alcácer Ceguer in 1450s and Arzila and Tangier around 1470. This solidified their rule over the southern part of the straits, and engaged in consistent deliberations whether to take Gibraltar, before Castile conquered it in 1462. They were mainly stopped because Castile had a "right of conqeust" of Gibraltar and Granada as a whole, and while Portuguese were seriously weighing to ignore it at times, they ultimately never attempted. If they tried, Castile might declare war on them, and Gibraltars position would be untenable. Portuguese involvement on Gibraltar straits ended in 17th century. Following Portuguese War of Independance, Ceuta opted to remain with the Spanish, while Tangier was ceded to British as dowry of Catharine Braganza in 1661.
Ultimately, Portuguese did control parts of the straits, on the African side from 15th to 17th century. The never really had a chance for taking european side as they were thwarted by Castile-Leonese conquests before they had the opportunity to even try
