Before we get too crazy, let's consider for a second the importance of 12 in units of measure. It shows up in time: minute, second, length (imperial ft), and I'm sure many other places.
The benefit is it's evenly divisible by 2, 3, and 4. How would we define seasons with 13 months?
Not even, dozenalists have pointed out that you can do base 12 on your fingers by counting your joints, 3 in each finger with your thumb as the pointer, you can even do it on both hands and get all the way to "100"
I don't see the problem, really. Base twelve is neat and all, but we're not directly going to do a lot of math with the number of months in a year, are we? And who says all seasons have to be equally long, or be defined in terms of whole months in the first place? Most people I know don't define seasons directly on the months, since they vary from year to year.
Personally I think you're looking at the wrong direction, it's hard to keep mental track of because the calendar doesn't reflect the time of seasonal change in most places where that matters. Doesn't have to be the exact equinox but roughly saying "the first day of the year marks spring" and "October 1st means the start of fall" would probably help folks remember seasons more easily.