Ok, I see why you wrote "picks the scale".The scale of the electroweak Lagrangian is set by the parameters that describe the Higgs potenial, namely μ and λ. These, along with the weak coupling constant g, determine the masses for the Higgs boson and the gauge bosons, all around 100 GeV. This is the electroweak scale, or sometimes called the Fermi scale.
Notice that his result is only one contribution to the vacuum energy. There are other contributions, equally important. With this approach, the puzzle is how the remaining contributions can nearly cancel this one.
The case of the Higgs model is however special in that a constant appears in the lagrangian. It is not an infinity due to vacuum fluctuations. I would therefore hesitate to lump it up with the vacuum catastrophe.