If the facility was in any way at fault in causing the damage--e.g. the gate was improperly maintained; or one of their employees closed it on you--then they are liable for any costs, damage, etc. it caused. (Conversely, if they weren't at fault--for example, if the gate was working correctly, but it closed on you because you waited until it was starting to shut to drive through it--then they are not liable.) If you believe they were at fault but they (or their insurer) will not voluntarily pay, you could sue them for the money. You would sue the facility itself; you do not sue their insurer.