Yes, but there are usually better options. If the at-fault driver has insurance, you should be able to collect from that insurance policy. In cases of serious injury, you may also be able to pursue compensation from the at-fault driver's umbrella or excess liability policies. If the at-fault driver does not have enough insurance to pay for all your medical bills and other losses, you can file what is known as an "underinsured motorist" (UIM) claim with your own insurance company. You might also be able to "stack" your UIM policies.
So in sum, you can go after the at-fault driver's assets, but start with the insurance policies.