A good friend of mine who has been growing tomatoes for many years told me to do 3 things: "Mulch, mulch, mulch!"
Mulching means putting down a layer of "stuff" on the ground around your plants. This is done to keep your soil evenly moist by preventing evaporation, to keep the dirt from splashing on your plants during rain, and to keep weeds from growing in your garden. This "stuff" can be a variety of organic material like leaves, grass, hay, wood chips, and so on. I use grass because it's readily available every time I mow the lawn and at the end of the season I can just leave what remains of it on top of my beds (to keep weeds from growing and also to enrich the soil) and turn it over the following spring or throw it into the composter if there's too much left to turn over.
After the soil warms up with the heat in June I cover my raised beds with a layer of about 2-3 inches of cut grass, but I leave an unmulched area (about a 6-9 inch diameter) around the stems. I don't think it's a good idea to have the grass touching the stem, and leaving an unmulched area around the stem makes it much easier to water. Throughout the season, my mulch disintegrates little by little, so I may remulch once or twice a season.
If you decide to use grass to mulch your tomato plants, be aware that if you use freshly-cut grass you will have an interesting odor in your garden for a few days. One solution would be to dry the grass for a few days first, but the smell really doesn't bother me so I just go ahead and use it as I cut it.