The ideals of courage, chivalry, and heroism depicted in Beowulf and Sir Gawain and the Green Knight still influence British culture to some extent, though the influence has been steadily lessening since the 1950s. However, it is far from clear that these ideals actually derive from Old English or Middle English poetry rather than classical sources such as Homer and Virgil. The Latin and (since the Renaissance) Greek classics have been a far more important element in British education than Old and Middle English texts. Though the influence of the classics has waned in recent decades, so has the teaching of pre-Shakespearean English literature. It is now possible to study English literature at a British university without having read anything written before 1590.
Insofar as Beowulf has any influence, it is probably in the continued importance of physical courage in an age where the need for it has largely died out. Beowulf is, above all, courageous. He actively seeks out adventure and the opportunity for heroism as a young man. Then, when he is king, he protects his people until he can no longer do so, when he dies in the attempt. Such heroism in battle is still an important part of British culture, though it is principally appreciated from a distance. For example, this valuation of military valor can be seen in the way the Second World War is obsessively revisited in films, television programs, and the media.
The ideals of chivalry and resistance to temptation in Sir Gawain and the Green Knight are also influential. This is perhaps most commonly seen in frequent tabloid scandals about politicians like Lord Sewel or Lord Archer succumbing to various temptations such as sex, drugs, or bribery. The implied standard to which they are held is seldom mentioned, but it is clearly related to Sir Gawain's code of chivalry, which includes a different type of strength from Beowulf's: purity in the face of temptation. The fact that the British honors system means that the subjects of these scandals often carry the same noble titles as Arthurian knights and lords makes the comparison even more appropriate.