Your permanent tax residence is the state which will expect you to file a Resident tax return. This means that all your income will be reported to Connecticut, and you will file Non-Resident tax returns for the states in which you worked reporting your income earned only for that state.
The U.S. Supreme Court ruled against double taxation in 2015, stating that two or more states are no longer permitted to tax the same earnings. So, if you live in Connecticut but work in New York and pay taxes on that income in New York, Connecticut cannot tax you on it as well. Connecticut must offer you a credit for taxes you paid elsewhere.
So you won't pay taxes on the some money twice. You'll have to spend a little more time preparing multiple state returns, however.