The Gulf was always going to be important to the two superpowers. For one thing it had oil and though neither America nor Russia was dependent on it, their allies were, whilst for capitalism, the markets were too. So any threat to the supply of oil, or any hike in its price, was a cause for concern.
And the withdrawal of Britain and France from their imperial presence in the region had been replaced by a growing American presence and challenged by a growing Soviet presence. For America and the West, stability in the region was important too. There was Israel and there was the Suez Canal. As for the Soviet Union, Islam was a threat as its central Asian states bordering the region (Iran had a long border with the Soviet Union) were Islamic and they were concerned lest Islam encourage a nationalist push to breakaway. Both superpowers would get their fingers burnt: the Soviet Union in Afghanistan, America in Iran. As for the Iran-Iraq War, both superpowers were in the same fix: if Iran won, the Islamic Revolution would spread too; if Iraq won, Saddam Hussein could come to dominate the whole region; whilst for America there was the added problem that, with increased power, either could come to threaten Israel. But neither superpower would want to see the other gain an advantage in any region.
So as with any region, even the Arctic and Antarctic, the superpowers had their own special, or else shared, strategic interests. Anything that happened was of interest to them and you will see their presence as the Gulf went to war with itself.