It's my understanding that teaching career sailors to swim is a relatively modern innovation - at best, there just wasn't the time/money for it...and at worst, you don't want your sailors to get the idea that they can just dive in the water and swim away from a fight gone sour. Does this hold true for the Scandinavians of the Viking Age?
I do not know of any source that mentions teaching Viking Age sailors to swim, but there are definitely other references to swimming.
Grettir's Saga has a lot of swimming, for instance:
Ch 29:
Grettir ducked beneath the flank of his horse and drove his stick into Odd's side with such violence that three of his ribs were broken and Odd fell into the pool with his horse and all the mares that were tethered there by the bank. Some people swam out and rescued them. There was great excitement about it. Kormak's men on one side and those of Bjarg on the other seized their arms, but the men of Hrutafjord and Vatnsnes came between them and parted them.
Ch 38:
Then Grettir threw off his clothes and got ready to go into the water. He went in a cloak and breeches of coarse stuff. He tucked up the cloak, tied a cord of bast round his waist, and took a barrel with him. Then he jumped overboard, swam across the channel and reached the land on the other side. There he saw a house standing and heard sounds of talking and merriment issuing from it. So he went towards the house.
Ch. 56:
Then he threw off his clothes and his weapons and swam out to the nets. He gathered them together, returned to the shore and cast them up on to the bank. Just as he was about to land Thorir quickly seized his short sword and drew it. He ran towards Grettir as he stepped on to the bank and aimed a blow at him. Grettir threw himself down backwards into the water and sank like a stone. Thorir stood by the shore intending to guard it until he came up. Grettir swam beneath the water, keeping close to the bank so that Thorir could not see him, and so reached the bay behind him, where he landed without letting himself be seen. The first Thorir knew of it was when Grettir lifted him up over his head and dashed him down with such violence that the sword fell out of his hand. Grettir got possession of it and without speaking a word cut off his head. So his life ended. After that Grettir refused to take in any forest-men, and yet he could not live alone.
Ch 58:
Grettir and Bjorn swam in one course the whole length of the Hitara from the lake at its head down to the sea.
Ch 59:
Directly he got his feet under him again he ran off to a large pool and swam across the river. In the evening he reached the settlement called Hrossholt, very exhausted.
Ch 66:
Grettir struck few more blows at the giant before he was dead. He then entered the cave, kindled a light and explored. It is not told how much treasure he found there, but there is supposed to have been some. He stayed there till late into the night and found the bones of two men, which he carried away in a skin. Then he came out of the cave, swam to the rope and shook it, thinking the priest was there; finding him gone he had to swarm up the rope and so reached the top.
Ch74-75, Grettir swims from an island off the coast to the mainland Iceland
Grettir then prepared for his swim. He wore a cloak of coarse material with breeches and had his fingers webbed. The weather was fine; he left the island towards the evening. Illugi thought his journey was hopeless. Grettir had the current with him and it was calm as he swam towards the fjord. He smote the water bravely and reached Reykjanes after sunset.
Heimskringla (among other sagas) mentions swimming as a sport and competition
Sigurd: "Do you remember that I could drag you under water, when we swam together, as often as I pleased?" Eystein: "But I could swim as far as you, and could dive as well as you; and I could run upon snow-skates so well that nobody could beat me, and you could no more do it than an ox."
Later in Heimskringla:
It was a day of beautiful weather and warm sunshine, and many went out to swim, both from the long-ship and the merchant vessel. An Iceland man, who was among the swimmers, amused himself by drawing those under water who could not swim so well as himself; and at that the spectators laughed. When King Sigurd saw and heard this, he cast off his clothes, sprang into the water, and swam to the Icelander, seized him, and pressed him under the water, and held him there; and as soon as the Icelander came up the king pressed him down again, and thus the one time after the other.
There are also mythological mentions of swimming competitions, so, it seems to be a respected attribute among the strong.
Written sources from the end of the Viking era frequently have references to swimming, but never portray as as a sailor safety training.
There is plenty of routine mentions in character introductions where is on of the items in a list of skills, intellectual or athletic. Also, a few extraordinary swims are mentioned, such as Grettir swimming from the small Island of Drangey to the shores of Iceland.
I don´t really think we know how common swimming was among the general population, but it definately was a part of viking culture. At some point the tradition disappears.
It´s worth keeping in mind that the the water in the north would frequently be fairly cold and swimming would not only have been about technique or speed, but also endurance and building up an ability to resist the cold, as coldwater swimmers do today.