Most of the Connecticut River between Canaan and Colebrook, and south of North Stratford, NH, has enough water to paddle all season.

Upstream of Canaan, look for 500-1000 cfs on the Pittsburg Gage. Between Colebrook and North Stratford, you may hit bottom with a loaded boat in a few places when there is less than 500 cfs on the North Stratford Gage, especially below Lyman Falls.

Use these gauge heights to make informed decisions regarding whether conditions are too high, or if there is adequate water to paddle the upper most reaches. There are a couple places to check current water levels:

The National Weather Service's Advanced Hydrologic Prediction Service CT River Predictions Center which indicates flood levels and predictions for the next few days,

and

The United States Geologic Service Gages:
Pittsburgh, NH
North Stratford, NH
Dalton, NH
Wells River, NH
North Hartland, NH
North Walpole, NH
Montague, MA
Hartford, CT

Finally, Great River Hydro provides flow forecasts for areas below their dams at:
http://www.h2oline.com/default.aspx?pg=co&op=TRC