You have exactly the same problem with nuclear. At night you throw away power. So far wind turbines are only throwing away less than 1% of the power they produce.
The solution to large scale wind power generation is to first and foremost accept a bit of power wasted. After that there are a lot of options for reducing the waste: HVDC lines, hydro power (luckily wind power generates the most in winter when hydro power is at risk of running dry), load shifting... If waste becomes unacceptably high, those solutions become comparably more attractive.
I'm sorry, I probably emphasized the wrong point in my original comment. The real issue is not overproduction, but rather not producing the power when it is needed. There always needs to be a backup system available that can provide at least as much power as the wind turbines are providing to the grid. That basically means either having other types of power plants available to balance the load, or storing the excess energy that the wind turbines generate, which can then be used to compensate for lack of wind later on.
The problem with having power plants as backups is that most types are not brought online fast enough if they are completely shut down. The best type of load-balancing as far as I understand is hydroelectric, because you can just turn turbines off or on as the demand changes. Unfortunately there are only so many rivers that can be harnessed for that purpose. Any other types of power plants need to be kept more or less online to be able to respond to quick changes in demand.
As far as energy storage is concerned, the amount of energy required to compensate for a lack of wind power for even a few days is extremely high. With current technology the amount of resources needed, energy density, and the efficiency of different solutions (hydrogen, pumping water to a reservoir, large batteries etc.) is not very promising. Any kind of intermittent power generation, be it solar, wind or tidal, requires this kind of backup system and until large-scale energy storage technology gets more mature, any kind of large-scale implementation of those power-generating methods is a waste of time, money and energy that would be better spent on other things.
Having said that, I'm just wondering what could be achieved if the money spent for implementation would go towards research instead.