Well, one potential problem for swimming in the ocean here at the end of May is how do you find any open water! Normally (which means it could be different, but that would be exceedingly odd) there is shore ice for at least two or three miles out from Barrow until late in June.
And that ice is commonly not safe to walk on at that time etiher, so it isn't likely that you'd be able to find some convenient place and just hike out to where there is open water. Slipping through a crack on the way out could be serious. Regardless, getting totally wet a couple miles out on the ice is pretty serious even when intentional.
The best bet is to wait until mid-July, or at least late June. Around the first of August would be a very good time. There will almost certainly be lots of open water well into November, and probably in December too (there's a bit of shore ice today, but finding a place where with access wouldn't be too hard).
Keep in mind that the water temperature is in the range of 33F to 36F, and in late May that might be the high temperature for the air during the day too. It is unlikely to get up to 40F in May or the first week in June, but it could.