Should I use traditional 2x4 framing or 1x3 grid framing for a New England basement remodel?
You can't frame out in 1x3 especially in a basement.  It is just too flimsy and and warping with probably make it brittle.  If you want to save an inch or two (which you will never notice) then please frame in 2x3.  You won't have any issues with 2x3s (other than having to install shallow electrical boxes in some cases).  

Note:  Watched the video on the grid system.  I am sorry but this just seems very ill advised to me.  I have a list of problems I have with it (these are just some off the top of my head):


it's not faster or cheaper than installing traditional 2x3 (2x4) walls
it saves you maybe 1/2 inch per wall.
my biggest peeve is that you are drilling tons of holes in your foundation.  I just can't believe someone would do this
your insulation is now a support structure since your 1x3 will bend and rest on it
not even sure if running romex behind would be code (doubt it would get passed in my area)
basement walls usually aren't flat (and if they are they don't usually stay flat).  So when framing I pop out at the bulge you would just be drywalling there.