Today there are people on the Alaskan and Siberian sides of the strait who have regular contact and cultural exchange. Was this also the case in ancient times, more or less continuously since the population of the Americas? Or was it established more recently?
Yes, there has been extensive and continuous contact across the Bering Strait as far back as the archaeological record goes. Dorothy Jean Ray's The Eskimos of Bering Strait, 1650-1898 deals with the historical period of this cross-strait contact, but we know thanks to genetic, linguistic and archaeological research that contact has been maintained for thousands, if not tens of thousands of years. It is entirely possible that an unbroken chain of contact may stretch back to before the previous ice age.
The issue for modern scholarship to determine is how far that cross-strait contact spread. In 2011, a portion of a cast bronze buckle-like object was uncovered in an 11th-century home excavated at Cape Espenberg in northwest Alaska. It's the first bronze object dated to Alaska's prehistoric period, and radiocarbon dating of a piece of leather wrapped around the buckle-like object turned up a domain of about 600 AD.
Was this a relic of hand-to-hand trading from east Asia, or was it the remnant of more reliable trading routes? Does the mythical Strait of Anian have its origins with fourth-hand accounts of the Bering Strait?
We don't know.