Wire mesh is a good idea in that it maintains the same level of ventilation offered by the current openings. Only you can decide of the ventilation is necessary or desired. I could see problems of blowing and drifting snow getting inside the shed in the winter time and if that is a problem then you may be up to eliminating the openings.
If you do decide to go the route of using the wire mesh then the best method of fastening cut and fitted pieces in place is to use staples. These come in a variety of forms depending upon how they are applied. There are staples that can be installed with a hammer which may be the simplest as you may very well already have a hammer. Many sizes are available. For mesh you will want ones that are fairly small with legs of say 1/2 to 5/8 inches long.
Another type of staple is applied using a staple gun. These can be purchased fairly inexpensive and the staple gun for the type of staples may look something like below. The staples best to use for this application would have rounded tops and have legs 9/16 inch in length. Sometimes a staple gun like these will not sink the staple fully and if this happens just tap it in the rest of the way with a hammer.
The third choice is to use a pneumatic brad nailer that is able to use staples. These are great because they shoot the staples in quickly and with enough force that they will be sunk way in. One slight disadvantage with many brad nailers with staples is that the staples are narrow and the nose of the gun does not typically have a guide to line up the staple over the wire of the mesh. So until you get the eye for it you may get some staples that do not bridge across the wires of the mesh.
photos show examples only and are not meant to imply endorsement for any particular product