First, look at the label and see the manufacturer's recommendations.
You can try with a steam cleaner, or different recipes for homemade mattress cleaners.
If a blood stain appears, then treat it with some hydrogen peroxide, but keep in mind that hydrogen peroxide has bleaching properties and may whiten the fabric. Apply it on the stain and blot, repeat again until the stain is gone.
You can use this method on urine stains as well. Vinegar also fights against urine stains. The method is the same. After removing the stain, cover the area with baking soda to neutralize the odour that may occur.
A rule of thumb: do not allow the inner padding of the mattress to go dampen, or it will become extremely difficult for the mattress to get dry completely. This usually leads to mould and mildew growth.