First you’ll want to get rid of any mice currently living in the insulation. You can call in an exterminator or use traps to catch the mice nesting there already. If the basement is not frequented by family pets or children, you can also use poison. Remove any sources of food and water for the mice by thoroughly cleaning the basement and sealing up any food items stored there. If you suspect that food has been contaminated by rodents, throw it out. Make sure that faucets and taps are shut off so that mice don’t have access to water. Get rid of any debris in the basement that the mice can use to live in, such as old appliances, building materials, trash or recyclables.
One option for covering the rafters is with tight-fitting sheet metal. Make sure the corners and edges overlap the rafters -- mice can fit through an opening as small as 1/4-inch. Check the edges and corners frequently to make sure they are tight. If there are small openings, fill them temporarily with steel wool and tighten them as soon as possible because steel wool will leave rust stains if it gets damp.
Another covering material you can use on rafters is plaster. Fill the spaces between the rafters with copper gauze then cover the entire area with plaster. Check the plaster frequently for any new holes or damage and refill and cover them right away. Concrete mortar and concrete patching mixes are also suitable for covering rafters.
You can also use rubber sheeting to cover the ceiling space. As with plaster, fill in any gaps between the existing insulation and the sheeting with copper gauze then attach the rubber sheeting to the rafters, making a tight seal against the edges and corners. As with other covers, check the rubber sheeting on a regular basis for wear and tear.