Can you fly a helicopter without a driver's license?
Here's a a video where a CFI discusses this very question. There's no rule that says you need a driver's license to obtain a certificate to fly in the US. The real question is why you can't obtain a driver's license. If you have a medical problem that disqualifies you from driving, that will likely pose a far larger issue getting a medical to fly. Similarly, any drug or alcohol problem or conviction must be reported to the FAA. If your driver's license was suspended or revoked due to drug or alcohol, that must be reported as well. These issues are reviewed on a case-by-case basis, as the FAA wants to be satisfied that you will not try to fly while impaired. Beyond those types of problems, if you simply don't own a car and never wanted to learn to drive, nothing is stopping you from getting a license to fly.

Note that you will need some sort of government-issued photo ID. A passport or state-issued non-driver ID should be fine. You might also run into the practical problem that it may be difficult to get to/from airports in many regions without driving, so you'd have to find alternate transportation.

Finally, there is one sort of connection between driver's licenses and flying in the US. Pilots with a valid driver's license can fly Light Sport Aircraft without obtaining the otherwise-required third class medical certificate. A few rotorcraft (like this one) qualify as LSA. There are, however, significant restrictions on these operations, such as the requirement that they can't fly at night or carry more than one passenger. This rule does not override the medical requirements to fly, but requires the pilot to certify him/herself able to fly safely instead of receiving that certification from a Aviation Medical Examiner. In other words, if there's a reason you wouldn't be able to obtain a third class medical, then you are not allowed to fly a LSA using only a driver's license.