How long has the contractor been working under the current name? Ideal for it to be at least 3-5 years as a business under the current name - not just "years of experience".
Does your work require a license?
Do they have an office? Working out of a garage or pick-up truck = here today, gone tomorrow.
Does the contractor carry insurance? General liability and worker's comp are a must. Provided they have worker's comp insurance, if a worker gets hurt on the job the property owner cannot be sued.
Does the contractor have their own employees, use sub-contractors or both? Beware of the suitcase contractor - a real contractor will at least have some employees of their own and use subs for specialty jobs.
Are there unanswered complaints with the BBB? Any business can have complaints against them. The good ones always answer regardless of the complaint.
Is your contractor willing to give you a reference list?
Will your contractor stay on the job once it starts or will they be bouncing around between projects?
Never sign the contract on the first visit. Don't let the contractor pressure you into signing by saying the price is only good for today. Sleep on it and if they won't honor the price the next day find another contractor.
Never pay upfront when you sign the contract. Clearly negotiate the schedule of payments with the first being when the work begins or when material are purchased with the final being due upon completion.