Our gable vent is falling apart. After reading some forums, We are not sure if we need to replace this as we had roof vents installed with our new roof. Could we just close it off or cover it with screening?
Roofers generally have no idea how to properly vent a roof, so it is doubtful your's was done correctly. They put one or two vents up there and they're done.
Since the more efficient way to accomplish roof venting is through ridge and soffit vents, you're roof is not the exception to the rule.
To check if the roof has the proper venting you would need a minimum of 1 sq inch of venting for every sq ft of attic space. If you have crawl spaces behind knee walls, include this in your figures. Each roof vent has the venting area on the vent. You can go to a store to get the vent figure if getting on the roof is not your thing. Compare the two figures and see what you come up with.
Does the house have soffit vents?