In New Haven
While traveling around the New Haven area, you should carry a copy/scan/picture of your I-94 record and your passport page with your biographic information and photo with you. Keep your physical passport and other immigration documents in a secure location at home.
Outside of New Haven
When traveling outside the New Haven area, carry your passport and immigration documents; they will be required for a domestic flight, but could also be requested on trips where you are traveling by car, train or bus. This is particularly relevant if you are traveling within two or three hours from the northern or southern U.S. borders. Your U.S. issued driver's license if generally not evidence of lawful status in the U.S.
While it is extremely unlikely you would be questioned, having these documents with you provides immediate proof of your lawful status in the U.S.