Answers

Depending on the type of soffit/eave vents, it would be easiest and most cost effective to replace them with a new vent that has smaller screen material.  Most of those can be found at a local supply store.

As previous answers point to adding proper insulation in the attic, it is also recommended that using a leak barrier under the roof shingles from the eave of the roof back until you reach 24 inches above an interior wall.  In other words, the leak barrier, a rubber like membrane that adheres to the roof deck, will cover the overhang and at least 24 inches above the interior wall of the house around the perimeter.  This membrane helps prevent water from melting ice dams from entering the home and overhang (soffit).  Its rubber like properties help seal around the nails that are driven through it that hold the shingles on.  If your roof does not currently have a leak barrier, the existing shingles will have to be removed and replaced where the leak barrier is to be installed.  A couple of recommended leak barrier products from GAF are WeatherWatch and StormGuard.