If there is snow, Eaglecrest is it for lift served alpine skiing close to Juneau. It's just a short drive across the bridge to Douglas Island and then up the one road North to Eaglecrest Road. You can't drive from Juneau to anywhere else as the only connection to anywhere else is via ferry, boat, or air service. Whitehorse, Yukon Territory may have a lift now, but the trip by ferry and car is considerable more than you specify. If you go South by ferry to Prince Rupert, BC and then drive the Trans-Canada Highway there are ski areas with some real possibilities. One near Terrace, BC, and one outside Smithers, BC. Quite a drive from Juneau though. Going to Haines for heli-skiing sounds good, but is hardly a day trip what with ferry schedules and weather restrictions.
When I lived in Juneau I had a pair of randonee skis I used wherever I could find snow. There could be feet of snow one day and then rain the next day would wash it away. I probably trespassed some here and there, but it was as close as I could get to skiing.
It's worth going to Juneau anyway.